Big Ideas Math Algebra 2 Chapter 9 Review

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Big Ideas Math Book Algebra 2 Answer Primal Chapter 2 Quadratic Functions

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• Quadratic Functions Maintaining Mathematical Proficiency – Folio 45
• Quadratic Functions Mathematical Practices – Page 46
• Lesson 2.1 Transformations of Quadratic Functions – Page (48-54)
• Transformations of Quadratic Functions 2.i Exercises – Folio (52-54)
• Lesson 2.ii Characteristics of Quadratic Functions – Page (56-64)
• Characteristics of Quadratic Functions 2.2 Exercises – Page (61-64)
• Quadratic Functions Report Skills Using the Features of Your Textbook to Fix for Quizzes and Tests – Page 65
• Quadratic Functions 2.one – ii.2 Quiz – Folio 66
• Lesson 2.iii Focus of a Parabola – Folio (68-74)
• Focus of a Parabola two.iii Exercises – Page (72-74)
• Lesson ii.4 Modeling with Quadratic Functions – Page (76-82)
• Modeling with Quadratic Functions 2.iv Exercises – Page (fourscore-82)
• Quadratic Functions Performance Task: Blow Reconstruction – Page 83
• Quadratic Functions Chapter Review – Page (84-86)
• Quadratic Functions Chapter Exam – Page 87
• Quadratic Functions Cumulative Assessment – Page (88-89)

Quadratic Functions Maintaining Mathematical Proficiency

Find the x-intercept of the graph of the linear equation.

Question 1.
y = 2x + 7

Question ii.
y = -6x + 8

Question 3.
y = -10x – 36

Question iv.
y = 3(10 – v)

Question five.
y = -4(x + 10)

Question vi.
3x + 6y = 24

Find the altitude between the two points.

Question vii.
(2, 5), (-4, 7)

Question 8.
(-i, 0), (-eight, iv)

Question ix.
(3, ten), (v, 9)

Question 10.
(7, -4), (-5, 0)

Question 11.
(iv, -viii), (4, two)

Question 12.
(0, nine), (-three, -6)

Question 13.
Abstract REASONING Use the Altitude Formula to write an expression for the distance between the two points (a, c) and (b, c). Is there an easier way to observe the distance when the x-coordinates are equal? Explain your reasoning

Quadratic Functions Mathematical Practices

Monitoring Progress

Determine whether the syllogism represents right or flawed reasoning. If flawed, explain why the decision is non valid.

Question 1.
All mammals are warm-blooded.
All dogs are mammals.
Therefore, all dogs are warm-blooded.

Question 2.
All mammals are warm-blooded.
My pet is warm-blooded.
Therefore, my pet is a mammal.

Question three.
If I am sick, then I will miss school.
I missed school.
Therefore, I am sick.

Question 4.
If I am ill, then I will miss schoolhouse.
I did not miss school.
Therefore, I am not ill.

Lesson ii.one Transformations of Quadratic Functions

Essential Question

How do the constants a, h, and k affect the graph of the quadratic function g(x) = a(x – h)² + k?
The parent function of the quadratic family is f(10) = x². A transformation of the graph of the parent function is represented by the role k(x) = a(x – h)² + k, where a ≠ 0.

EXPLORATION ane
Identifying Graphs of Quadratic Functions
Work with a partner. Match each quadratic function with its graph. Explicate your reasoning. Then apply a graphing calculator to verify that your answer is correct.
a. g(x) = -(x – two)^two
b. g(x) = (x – ii)² + 2
c. g(x) = -(x + 2)² – 2
d. g(x) = 0.v(ten – two)² + two
e. g(x) = 2(x – 2)²
f. thousand(ten) = -(10 + 2)^two + ii

Communicate Your Answer

Question two.
How do the constants a, h, and k affect the graph of the quadratic part g(10) =a(x – h)² + k?

Question three.
Write the equation of the quadratic function whose graph is shown at the right. Explain your reasoning. Then utilise a graphing calculator to verify that your equation is correct.

2.1 Lesson

Monitoring Progress

Describe the transformation of f(x) = 10² represented by g. And so graph each office.

Question 1.
thousand(x) = (10 – 3)ⁱⁱ

Question 2.
g(x) = (x + ii)^two – two

Question 3.
g(x) = (x + v)² + 1

Describe the transformation of f(x) = x² represented past g. And then graph each office.

Question 4.
g(x) = (\(\frac{1}{iii} x\))ⁱⁱ

Question 5.
one thousand(10) = three(x – 1)²

Question 6.
g(x) = -(x + 3)² + 2

Question 7.
Let the graph of grand be a vertical compress past a factor of \(\frac{ane}{2}\) followed by a translation 2 units up of the graph of f(10) = ten². Write a rule for g and place the vertex.

Question viii.
Allow the graph of k be a translation 4 units left followed by a horizontal shrink by a factor of \(\frac{1}{3}\) of the graph of f(x) = ten² + x. Write a rule for g.

Question 9.
WHAT IF? In Example v, the water hits the footing 10 feet closer to the burn truck after lowering the ladder. Write a function that models the new path of the water.

Transformations of Quadratic Functions 2.1 Exercises

Vocabulary and Core Concept Check

Question i.
Consummate THE SENTENCE The graph of a quadratic function is called a(n) ________.
Answer:

Question two.
VOCABULARY Identify the vertex of the parabola given past f(x) = (10 + 2)² – 4.
Respond:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, describe the transformation of f(x) = x^two represented by grand. Then graph each function.

Question 3.
thousand(x) = x² – 3
Reply:

Question 4.
g(ten) = x² + 1
Answer:

Question 5.
g(x) = (ten + 2)²

Reply:

Question half dozen.
chiliad(x) = (10 – 4)^two

Answer:

Question 7.
m(x) = (x – i)²

Answer:

Question eight.
yard(x) = (x + 3)²

Reply:

Question 9.
one thousand(x) = (ten + half dozen)^two – 2
Answer:

Question 10.
g(x) = (x – 9)² + 5
Respond:

Question 11.
g(10) = (10 – 7)ⁱⁱ + one
Reply:

Question 12.
g(x) = (x + x)ⁱⁱ – iii
Respond:

ANALYZING RELATIONSHIPS In Exercises 13–16, match the function with the correct transformation of the graph of f. Explain your reasoning.

Question xiii.
y = f(ten – ane)
Answer:

Question 14.
y = f(x) + ane
Answer:

Question 15.
y = f(x – i) + i
Answer:

Question 16.
y = f(x + i)
Respond:

In Exercises 17–24, depict the transformation of f(ten) = xⁱⁱ represented by thou. Then graph each role.

Question 17.
g(10) = -ten²

Answer:

Question 18.
g(x) = (-x)²

Respond:

Question 19.
g(x) = 3x²

Reply:

Question xx.
thousand(x) = \(\frac{1}{3}\)x²

Answer:

Question 21.
m(x) = (2x)²
Respond:

Question 22.
thousand(ten) = -(2x)²

Answer:

Question 23.
g(x) = \(\frac{i}{5}\)x^two – iv
Answer:

Question 24.
g(x) = \(\frac{1}{ii}\)(x – 1)²

Answer:

ERROR ANALYSIS In Exercises 25 and 26, depict and correct the error in analyzing the graph of f(x) = −6xⁱⁱ + 4.

Question 25.

Answer:

Question 26.

Respond:

USING STRUCTURE In Exercises 27–30, describe the transformation of the graph of the parent quadratic function. Then identify the vertex.

Question 27.
f(x) = iii(x + ii)ⁱⁱ + 1
Answer:

Question 28.
f(10) = -4(10 + 1)² – v
Answer:

Question 29.
f(x) = -2x^two + 5
Reply:

Question xxx.
f(x) = \(\frac{1}{2}\)(x – 1)²
Answer:

In Exercises 31–34, write a rule for g described by the transformations of the graph of f. And then place the vertex.

Question 31.
f(ten) = x² vertical stretch by a factor of 4 and a reflection in the x-axis, followed by a translation ii units up
Reply:

Question 32.
f(x) = x²; vertical shrink by a factor of \(\frac{one}{3}\) and a reflection in the y-axis, followed by a translation 3 units right
Reply:

Question 33.
f(x) = 8x² – half dozen; horizontal stretch by a factor of 2 and a translation 2 units up, followed past a reflection in the y-axis
Reply:

Question 34.
f(x) = (10 + half dozen)² + three; horizontal shrink by a factor of \(\frac{1}{2}\) and a translation 1 unit downwardly, followed by a reflection in the x-axis
Answer:

USING TOOLS In Exercises 35–40, match the function with its graph. Explicate your reasoning.

Question 35.
g(x) = 2(10 – 1)^two – 2
Reply:

Question 36.
g(10) = \(\frac{i}{2}\)(x + ane)² – 2
Reply:

Question 37.
g(10) = -ii(x – 1)^two + 2
Answer:

Question 38.
1000(x) = ii(x + one)² + two
Answer:

Question 39.
g(ten) = -2(ten + 1)ⁱⁱ – 2
Answer:

Question forty.
1000(x) = two(x – 1)^two + two
Answer:

JUSTIFYING STEPS In Exercises 41 and 42, justify eachstep in writing a function yard based on the transformationsof f(ten) = 2xⁱⁱ + 6x.

Question 41.
translation 6 units down followed past a reflection in the ten-axis

Respond:

Question 42.
reflection in the y-centrality followed past a translation four units right

Answer:

Question 43.
MODELING WITH MATHEMATICS The function h(10) = -0.03(x – fourteen)ⁱⁱ + six models the jump of a red kangaroo, where x is the horizontal distance traveled (in feet) and h(x) is the height (in feet). When the kangaroo jumps from a college location, it lands 5 feet further abroad. Write a office that models the second leap.

Respond:

Question 44.
MODELING WITH MATHEMATICS The function f(t) = -16tⁱⁱ + 10 models the height (in feet) of an object t seconds after it is dropped from a height of 10 feet on Earth. The same object dropped from the same pinnacle on the moon is modeled by chiliad(t) = –\(\frac{8}{3}\)t² + 10. Describe the transformation of the graph of f to obtain yard. From what summit must the object be dropped on the moon so information technology hits the ground at the aforementioned time as on Earth?
Answer:

Question 45.
MODELING WITH MATHEMATICS Flying fish apply their pectoral fins like airplane wings to glide through the air.
a. Write an equation of the form y = a(x – h)^two + k with vertex (33, v) that models the flying path, assuming the fish leaves the water at (0, 0).
b. What are the domain and range of the function? What do they correspond in this situation?
c. Does the value of a modify when the flight path has vertex (xxx, 4)? Justify your answer.

Answer:

Question 46.
HOW DO YOU See Information technology? Describe the graph of g as a transformation of the graph of f(ten) = x².

Answer:

Question 47.
COMPARING METHODS Let the graph of chiliad be a translation 3 units up and 1 unit right followed by a vertical stretch by a cistron of 2 of the graph of f(x) = 10².
a. Identify the values of a, h, and thou and utilise vertex course to write the transformed function.
b. Use office notation to write the transformed function. Compare this role with your function in office (a).
c. Suppose the vertical stretch was performed commencement, followed by the translations. Repeat parts (a) and (b).
d. Which method exercise you prefer when writing a transformed function? Explain.
Answer:

Question 48.
Thought PROVOKING A jump on a pogo stick with a conventional bound can be modeled by f(ten) = -0.v(10 – 6)^two + 18, where x is the horizontal distance (in inches) and f(x) is the vertical distance (in inches). Write at least 1 transformation of the role and provide a possible reason for your transformation.
Answer:

Question 49.
MATHEMATICAL CONNECTIONS The expanse of a circumvolve depends on the radius, as shown in the graph. A round earring with a radius of r millimeters has a round hole with a radius of \(\frac{3 r}{four}\) millimeters. Describe a transformation of the graph below that models the area of the bluish portion of the earring.

Answer:

Maintaining Mathematical Proficiency
A line of symmetry for the figure is shown in red. Notice the coordinates of bespeak A. (Skills Review Handbook)

Question 50.

Answer:

Question 51.

Answer:

Question 52.

Answer:

Lesson 2.ii Characteristics of Quadratic Functions

Essential Question
What type of symmetry does the graph of f(x) = a(x – h)² + g have and how can yous draw this symmetry?

EXPLORATION 1
Parabolas and Symmetry
Piece of work with a partner.
a. Complete the table. Then utilize the values in the table to sketch the graph of the role
f(10) = \(\frac{1}{2}\)tenⁱⁱ – 2x – 2 on graph paper.


b. Utilise the results in role (a) to identify the vertex of the parabola.
c. Find a vertical line on your graph paper so that when you fold the newspaper, the left portion of the graph coincides with the right portion of the graph. What is the equation of this line? How does it relate to the vertex?

d. Show that the vertex form f(ten) = \(\frac{ane}{2}\)(ten – 2)² – 4 is equivalent to the office given in part (a).

EXPLORATION 2
Parabolas and Symmetry
Work with a partner. Repeat Exploration 1 for the function given past f(ten) = –\(\frac{ane}{3}\)xⁱⁱ + 2x + 3 = –\(\frac{1}{3}\)(x – 3),sup>2 + half-dozen.

Communicate Your Answer

Question 3.

What type of symmetry does the graph of f(10) = a(10 – h)² + chiliad have and how tin can you draw this symmetry?

Question 4.
Draw the symmetry of each graph. And so apply a graphing reckoner to verify your respond.
a. f(x) = -(x – 1)² + 4
b. f(x) = (x + ane)² – two
c. f(x) = 2(x – 3)² + ane
d. f(ten) = \(\frac{1}{2}\)(10 + 2)²
e. f(ten) = -2xⁱⁱ + three
f. f(10) = 3(ten – 5)² + 2

2.ii Lesson

Monitoring Progress

Graph the function. Characterization the vertex and axis of symmetry.

Question 1.
f(ten) = -3(10 + ane)²

Question two.
grand(10) = 2(x – 2)² + 5

Question 3.
h(x) = 10ⁱⁱ + 2x – one

Question 4.
p(x) = -2xⁱⁱ – 8x + i

Question 5.
Discover the minimum value or maximum value of
(a) f(x) = 4x² + 16x – 3 and
(b) h(x) = -x^two + 5x + nine. Describe the domain and range of each part, and where each role is increasing and decreasing.

Graph the function. Label the x-intercepts, vertex, and centrality of symmetry.

Question vi.
f(10) = -(x + 1)(10 + v)

Question 7.
1000(x) = \(\frac{1}{4}\)(x – vi)(x – 2)

Question 8.
WHAT IF? The graph of your third shot is a parabola through the origin that reaches a maximum height of 28 yards when x = 45. Compare the altitude it travels before it hits the ground with the distances of the first two shots.

Characteristics of Quadratic Functions two.2 Exercises

Vocabulary and Core Concept and Bank check

Question 1.
WRITING Explicate how to decide whether a quadratic function will take a minimum value or a maximum value.
Answer:

Question two.
WHICH 1 DOESN'T Vest? The graph of which function does non vest with the other three? Explain.

Answer:

Question iii.
f(10) = (x – 3)^two
Reply:

Question 4.
h(x) = (x + 4)²
Reply:

Question 5.
g(x) = (x + 3)² + v
Respond:

Question half dozen.
y = (ten – 7)² – 1
Reply:

Question 7.
y = -iv(x – 2)² + 4
Answer:

Question viii.
g(x) = 2(x + ane)² – 3
Respond:

Question 9.
f(x) = -2(x – i)² – 5
Answer:

Question 10.
h(x) = 4(x + 4)² + 6
Reply:

Question 11.
y = –\(\frac{1}{4}\)(ten + 2)² + i
Answer:

Question 12.
y = \(\frac{one}{two}\)(x – 3)² + 2
Answer:

Question 13.
f(x) = 0.4(x – 1)²
Answer:

Question 14.
thousand(x) = 0.75x^two – 5
Answer:

ANALYZING RELATIONSHIPS In Exercises 15–18, use the axis of symmetry to lucifer the equation with its graph.

Question 15.
y = 2(x – 3)ⁱⁱ + ane
Answer:

Question 16.
y = (x + iv)² – 2
Reply:

Question 17.
y = \(\frac{1}{2}\)(x + i)² + 3
Respond:

Question xviii.
y = (10 – 2)^two – one

Reply:

REASONING In Exercises 19 and 20, use the centrality of symmetry to plot the reflection of each point and complete the parabola.

Question 19.

Answer:

Question xx.

Answer:

In Exercises 21–xxx, graph the function. Label the vertex and axis of symmetry.

Question 21.
y = x^two + 2x + 1
Reply:

Question 22.
y = 3xⁱⁱ – 6x + 4
Reply:

Question 23.
y = -4x² + 8x + 2
Reply:

Question 24.
f(x) = -x² – 6x + iii
Answer:

Question 25.
g(x) = -ten² – 1
Answer:

Question 26.
f(x) = 6x^two – v
Answer:

Question 27.
g(x) = -1.5x² + 3x + two
Answer:

Question 28.
f(10) = 0.5xⁱⁱ + x – 3
Answer:

Question 29.
y = \(\frac{iii}{two}\)ten² – 3x + 6
Reply:

Question 30.
y = –\(\frac{v}{2}\)xⁱⁱ – 4x – 1
Answer:

Question 31.
WRITING 2 quadratic functions have graphs with vertices (2, 4) and (2, -3). Explain why yous tin can not utilize the axes of symmetry to distinguish between the two functions.
Respond:

Question 32.
WRITING A quadratic function is increasing to the left of x = ii and decreasing to the right of x = 2. Will the vertex be the highest or lowest indicate on the graph of the parabola? Explain.
Answer:

ERROR Analysis In Exercises 33 and 34, describe and correct the error in analyzing the graph of y = 4xⁱⁱ + 24x − 7.

Question 33.

Answer:

Question 34.

Reply:

MODELING WITH MATHEMATICS In Exercises 35 and 36, x is the horizontal altitude (in feet) and y is the vertical distance (in feet). Find and translate the coordinates of the vertex.

Question 35.
The path of a basketball thrown at an angle of 45° can exist modeled by y = -0.02xⁱⁱ + ten + 6.
Answer:

Question 36.
The path of a shot put released at an bending of 35° tin exist modeled by y = -0.01x² + 0.7x + half dozen.

Answer:

Question 37.
ANALYZING EQUATIONS The graph of which function has the same centrality of symmetry as the graph of y = ten² + 2x + ii?
A. y = 2x² + 2x + 2
B. y = -3x^two – 6x + 2
C. y = 10² – 2x + 2
D. y = -5x² + 10x + 23
Respond:

Question 38.
USING STRUCTURE Which function represents the widest parabola? Explain your reasoning.
A. y = two(10 + three)²
B. y = 10ⁱⁱ – v
C. y = 0.5(x – i)² + 1
D. y = -ten² + 6
Answer:

In Exercises 39–48, observe the minimum or maximum value of the function. Draw the domain and range of the part, and where the function is increasing and decreasing.

Question 39.
y = 6xⁱⁱ – 1
Respond:

Question twoscore.
y = 9x² + 7
Answer:

Question 41.
y = -x^two – 4x – ii
Answer:

Question 42.
g(10) = -3x^two – 6x + five
Answer:

Question 43.
f(10) = -2x² + 8x + 7
Respond:

Question 44.
chiliad(x) = 3x² + 18x – 5
Answer:

Question 45.
h(x) = 2x² – 12x
Answer:

Question 46.
h(10) = ten² – 4x
Answer:

Question 47.
y = \(\frac{1}{4}\)x² – 3x + two
Respond:

Question 48.
f(x) = \(\frac{three}{2}\)x^two + 6x + 4
Reply:

Question 49.
PROBLEM SOLVING The path of a diver is modeled by the office f(ten) = -9x^two + 9x + 1, where f(x) is the meridian of the diver (in meters) above the water and ten is the horizontal altitude (in meters) from the end of the diving board.
a. What is the height of the diving board?
b. What is the maximum elevation of the diver?
c. Describe where the diver is ascending and where the diver is descending.

Respond:

Question l.
Trouble SOLVING The engine torque y (in foot-pounds) of 1 model of car is given by y = -3.75x² + 23.2x + 38.8, where x is the speed (in thousands of revolutions per minute) of the engine.
a. Notice the engine speed that maximizes torque. What is the maximum torque?
b. Explicate what happens to the engine torque every bit the speed of the engine increases.
Answer:

MATHEMATICAL CONNECTIONS In Exercises 51 and 52, write an equation for the area of the figure. Then decide the maximum possible area of the figure.

Question 51.

Answer:

Question 52.

Answer:

In Exercises 53–60, graph the office. Label the 10-intercept(s), vertex, and axis of symmetry.

Question 53.
y = (10 + 3)(ten – 3)
Reply:

Question 54.
y = (10 + 1)(ten – 3)
Answer:

Question 55.
y = 3(10 + 2)(x + half dozen)
Answer:

Question 56.
f(x) = 2(10 – 5)(x – 1)
Respond:

Question 57.
g(x) = -x(ten + half dozen)
Answer:

Question 58.
y = -4x(ten + seven)
Reply:

Question 59.
f(10) = -2(x – iii)²
Answer:

Question 60.
y = 4(x – 7)²
Respond:

USING TOOLS In Exercises 61–64, identify the x-intercepts of the function and describe where the graph is increasing and decreasing. Use a graphing figurer to verify your answer.

Question 61.
f(x) = \(\frac{one}{two}\)(ten – 2)(ten + 6)
Respond:

Question 62.
y = \(\frac{3}{iv}\)(x + 1)(10 – iii)
Answer:

Question 63.
m(ten) = -four(ten – 4)(x – 2)
Reply:

Question 64.
h(x) = -5(x + 5)(x + one)
Respond:

Question 65.
MODELING WITH MATHEMATICS A soccer actor kicks a ball downfield. The elevation of the ball increases until it reaches a maximum peak of eight yards, 20 yards away from the actor. A second kick is modeled by y = x(0.iv – 0.008x). Which kick travels farther before hitting the footing? Which kick travels higher?

Answer:

Question 66.
MODELING WITH MATHEMATICS Although a football field appears to be flat, some are actually shaped like a parabola so that rain runs off to both sides. The cross department of a field tin be modeled past y = -0.000234x(x – 160), where x and y are measured in anxiety. What is the width of the field? What is the maximum pinnacle of the surface of the field?

Reply:

Question 67.
REASONING The points (2, iii) and (-iv, 2) lie on the graph of a quadratic office. Decide whether you can employ these points to find the centrality of symmetry. If not, explain. If then, write the equation of the axis of symmetry.
Answer:

Question 68.
Open up-Concluded Write two unlike quadratic functions in intercept form whose graphs have the centrality of symmetry x= 3.
Reply:

Question 69.
PROBLEM SOLVING An online music store sells about 4000 songs each twenty-four hour period when it charges $1 per song. For each $0.05 increase in price, about 80 fewer songs per mean solar day are sold. Use the verbal model and quadratic function to decide how much the store should charge per song to maximize daily revenue.

Respond:

Question 70.
Problem SOLVING An electronics store sells seventy digital cameras per calendar month at a cost of $320 each. For each $twenty decrease in price, about 5 more than cameras per month are sold. Use the verbal model and quadratic part to determine how much the shop should charge per camera to maximize monthly revenue.

Answer:

Question 71.
DRAWING CONCLUSIONS Compare the graphs of the three quadratic functions. What do you lot observe? Rewrite the functions f and g in standard form to justify your answer.
f(10) = (x + iii)(x + 1)
g(x) = (x + two)^two – 1
h(x) = x² + 4x + 3
Respond:

Question 72.
USING STRUCTURE Write the quadratic function f(x) = x² + 10 – 12 in intercept form. Graph the function. Label the x-intercepts, y-intercept, vertex, and axis of symmetry.
Answer:

Question 73.
PROBLEM SOLVING A woodland jumping mouse hops along a parabolic path given by y = -0.2x² + 1.3x, where x is the mouse'due south horizontal distance traveled (in anxiety) and y is the respective height (in feet). Tin can the mouse jump over a fence that is 3 feet high? Justify your reply.

Answer:

Question 74.
HOW Do Y'all See IT? Consider the graph of the role f(ten) = a(x – p)(ten – q).

a. What does f(\(\frac{p+q}{2}\)) stand for in the graph?
b. If a < 0, how does your answer in function (a) change? Explain.
Answer:

Question 75.
MODELING WITH MATHEMATICS The Gateshead Millennium Bridge spans the River Tyne. The arch of the span tin exist modeled by a parabola. The curvation reaches a maximum summit of 50 meters at a point roughly 63 meters across the river. Graph the bend of the arch. What are the domain and range? What practise they represent in this situation?

Reply:

Quadratic 76.
THOUGHT PROVOKING
You lot have 100 anxiety of fencing to enclose a rectangular garden. Draw 3 possible designs for the garden. Of these, which has the greatest area? Brand a theorize about the dimensions of the rectangular garden with the greatest possible area. Explicate your reasoning.
Answer:

Question 77.
MAKING AN ARGUMENT The point (1, 5) lies on the graph of a quadratic function with axis of symmetry x = -1. Your friend says the vertex could be the indicate (0, v). Is your friend correct? Explain.
Respond:

Question 78.
Critical THINKING Find the y-intercept in terms of a, p, and q for the quadratic function f(10) = a(x – p)(x – q).
Answer:

Question 79.
MODELING WITH MATHEMATICS A kernel of popcorn contains water that expands when the kernel is heated, causing it to pop. The equations beneath stand for the "popping book" y (in cubic centimeters per gram) of popcorn with moisture content x (as a percentage of the popcorn'due south weight).
Hot-air popping: y = -0.761(ten – 5.52)(x – 22.6)
Hot-oil popping:y = -0.652(x – five.35)(x – 21.8)

a. For hot-air popping, what moisture content maximizes popping volume? What is the maximum volume?
b. For hot-oil popping, what moisture content maximizes popping book? What is the maximum book?
c. Use a graphing calculator to graph both functions in the same coordinate plane. What are the domain and range of each function in this situation? Explain.
Answer:

Question 80.
ABSTRACT REASONING A function is written in intercept course with a > 0. What happens to the vertex of the graph as a increases? equally a approaches 0?
Reply:

Maintaining Mathematical Proficiency

Solve the equation. Check for extraneous solutions. (Skills Review Handbook)

Question 81.
3\(\sqrt{ten}\) – half-dozen = 0
Answer:

Question 82.
2\(\sqrt{x-4}\) – 2 = two
Respond:

Question 83.
\(\sqrt{5x}\) + v = 0
Respond:

Question 84.
\(\sqrt{3x+8}\) = \(\sqrt{x+4}\)
Answer:

Solve the proportion. (Skills Review Handbook)

Question 85.
\(\frac{i}{2}\) = \(\frac{x}{4}\)
Reply:

Question 86.
\(\frac{2}{3}\) = \(\frac{x}{9}\)
Respond:

Question 87.
\(\frac{-1}{4}\) = \(\frac{iii}{x}\)
Answer:

Question 88.
\(\frac{five}{2}\) =-\(\frac{20}{x}\)
Reply:

Quadratic Functions Study Skills Using the Features of Your Textbook to Gear up for Quizzes and Tests

Core Vocabulary

Core Concepts

Section 2.1

Section 2.2

Mathematical Practices

Question 1.
Why does the height you found in Exercise 44 on page 53 make sense in the context of the state of affairs?

Question 2.
How can you effectively communicate your preference in methods to others in Practice 47 on page 54?

Question 3.
How can you use technology to deepen your understanding of the concepts in Practice 79 on page 64?

Written report Skills
Using the Features of Your Textbook to Prepare for Quizzes and Tests

• Read and sympathise the core vocabulary and the contents of the Core Concept boxes.
• Review the Examples and the Monitoring Progress questions. Apply the tutorials at BigIdeasMath.com for boosted help.
• Review previously completed homework assignments.

Quadratic Functions ii.one – 2.2 Quiz

2.one – two.ii Quiz

Describe the transformation of f(x) = x^two represented by thousand. (Section 2.1)

Question 1.

Question 2.

Question 3.

Write a rule for thou and identify the vertex. (Section 2.one)

Question iv.
Let g be a translation 2 units up followed by a reflection in the 10-axis and a vertical stretch by a factor of 6 of the graph of f(x) = x².

Question 5.
Let m be a translation 1 unit left and 6 units down, followed past a vertical shrink by a gene of \(\frac{1}{two}\) of the graph of f(x) = 3(x + ii)ⁱⁱ.

Question 6.
Let g be a horizontal shrink by a factor of \(\frac{1}{4}\), followed by a translation 1 unit up and three units correct of the graph of f(x) = (2x + 1)² – 11.

Graph the office. Characterization the vertex and axis of symmetry. (Department ii.2)

Question vii.
f(x) = 2(10 – i)^two – v

Question 8.
h(ten) = 3x² + 6x – two

Question 9.
f(ten) = 7 – 8x – xⁱⁱ

Discover the 10-intercepts of the graph of the function. So draw where the function is increasing and decreasing.(Section 2.two)

Question ten.
one thousand(x) = -three(10 + ii)(ten + 4)

Question xi.
one thousand(x) = \(\frac{one}{2}\)(ten – 5)(10 + ane)

Question 12.
f(x) = 0.4x(10 – half-dozen)

Question 13.
A grasshopper tin can leap incredible distances, up to xx times its length. The peak (in inches) of the jump above the basis of a ane-inch-long grasshopper is given by h(ten) = –\(\frac{1}{xx}\)x² + ten, where 10 is the horizontal distance (in inches) of the bound. When the grasshopper jumps off a rock, it lands on the ground 2 inches farther. Write a role that models the new path of the jump. (Section 2.one)

Question 14.
A rider on a stranded lifeboat shoots a distress flare into the air. The meridian (in feet) of the flare above the water is given past f(t) = -16t(t – 8), where t is time (in seconds) since the flare was shot. The passenger shoots a second flare, whose path is modeled in the graph. Which flare travels higher? Which remains in the air longer? Justify your reply. (Section 2.2)

Lesson 2.3 Focus of a Parabola

Essential Question
What is the focus of a parabola?
EXPLORATION 1
Analyzing Satellite Dishes
Work with a partner. Vertical rays enter a satellite dish whose cantankerous section is a parabola. When the rays hit the parabola, they reverberate at the same angle at which they entered. (Encounter Ray 1 in the effigy.)
a. Draw the reflected rays so that they intersect the y-axis.
b. What practice the reflected rays accept in common?
c. The optimal location for the receiver of the satellite dish is at a indicate chosen the focus of the parabola. Determine the location of the focus. Explain why this makes sense in this situation.

EXPLORATION 2
Analyzing Spotlights

Piece of work with a partner. Beams of calorie-free are coming from the seedling in a spotlight, located at the focus of the parabola. When the beams hit the parabola, they reflect at the same angle at which they hit. (See Beam one in the effigy.) Depict the reflected beams. What do they have in common? Would you consider this to be the optimal outcome? Explain.

Communicate Your Answer

Question 3.
What is the focus of a parabola?

Question 4.
Describe some of the properties of the focus of a parabola.

2.iii Lesson

Monitoring Progress

Question one.
Use the Altitude Formula to write an equation of the parabola with focus F(0, -3) and directrix y = three.

Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation.

Question two.
y = 0.5x²

Question 3.
-y = x²

Question 4.
y² = 6x

Write an equation of the parabola with vertex at (0, 0) and the given directrix or focus.

Question 5.
directrix: x = -three

Question half-dozen.
focus: (-ii, 0)

Question 7.
focus: (0, \(\frac{3}{2}\))

Monitoring Progress

Question viii.
Write an equation of a parabola with vertex (-i, 4) and focus (-1, 2).

Question nine.
A parabolic microwave antenna is xvi feet in bore. Write an equation that represents the cross section of the antenna with its vertex at (0, 0) and its focus 10 anxiety to the right of the vertex. What is the depth of the antenna?

Focus of a Parabola 2.iii Exercises

Vocabulary and Core Concept Cheque

Question one.
Consummate THE SENTENCE A parabola is the set of all points in a plane equidistant from a fixed point called the ______ and a stock-still line called the __________ .
Reply:

Question ii.
WRITING Explain how to find the coordinates of the focus of a parabola with vertex (0, 0)and directrix y = 5.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–x, use the Distance Formula to write an equation of the parabola.

Question 3.

Answer:

Question 4.

Reply:

Question five.
focus: (0, -2)
directrix: y = two
Answer:

Question half dozen.
directrix: y = 7
focus: (0, -7)
Reply:

Question 7.
vertex: (0, 0)
directrix: y = -half-dozen
Reply:

Question eight.
vertex: (0, 0)
focus: (0, 5)
Answer:

Question 9.
vertex: (0, 0)
focus: (0, -10)
Answer:

Question x.
vertex: (0, 0)
directrix: y = -9
Answer:

Question 11.
ANALYZING RELATIONSHIPS Which of the given characteristics describe parabolas that open down? Explicate your reasoning.
A. focus: (0, -6)
directrix: y = 6
B. focus: (0, -2)
directrix: y = 2
C.focus: (0, half dozen)
directrix: y = -6
D. focus: (0, -1)
directrix: y = 1
Answer:

Question 12.
REASONING Which of the following are possible coordinates of the point P in the graph shown? Explain.

A. (-6, -1)
B. (3, –\(\frac{one}{4}\))
C. (4, –\(\frac{iv}{9}\))
D. (i, –\(\frac{1}{36}\))
Eastward. (6, -1)
F. (2, –\(\frac{1}{18}\))
Respond:

In Exercises 13–20, identify the focus, directrix, and axis of symmetry of the parabola. Graph the equation.

Question thirteen.
y = \(\frac{1}{8}\)xⁱⁱ
Answer:

Question xiv.
y = –\(\frac{1}{12}\)x²
Answer:

Question 15.
x = –\(\frac{one}{20}\)y²
Answer:

Question sixteen.
x= \(\frac{one}{24}\)y²
Answer:

Question 17.
y² = 16x
Answer:

Question 18.
-10^two = 48y
Respond:

Question xix.
6x² + 3y = 0
Answer:

Question twenty.
8x^two – y = 0
Answer:

Error ANALYSIS In Exercises 21 and 22, depict and right the fault in graphing the parabola.

Question 21.

Answer:

Question 22.

Answer:

Question 23.
ANALYZING EQUATIONS The cross department (with units in inches) of a parabolic satellite dish can be modeled by the equation y = \(\frac{ane}{38}\)ten². How far is the receiver from the vertex of the cross section? Explain.
Answer:

Question 24.
ANALYZING EQUATIONS The cantankerous department (with units in inches) of a parabolic spotlight can be modeled by the equation x = \(\frac{1}{20}\)y². How far is the bulb from the vertex of the cross section? Explain.

Answer:

In Exercises 25–28, write an equation of the parabola shown.

Question 25.

Answer:

Question 26.

Reply:

Question 27.

Answer:

Question 28.

Answer:

In Exercises 29–36, write an equation of the parabola with the given characteristics.

Question 29.
focus: (3, 0)
directrix: ten = -3
Answer:

Question thirty.
focus: (\(\frac{2}{3}\), 0)
directrix: 10 = –\(\frac{2}{3}\)
Respond:

Question 31.
directrix: ten = -10
vertex: (0, 0)
Answer:

Question 32.
directrix: y = \(\frac{8}{3}\)
vertex: (0, 0)
Answer:

Question 33.
focus: (0, –\(\frac{5}{three}\))
directrix: y = \(\frac{5}{3}\)
Respond:

Question 34.
focus: (0, \(\frac{5}{4}\))
directrix: y = –\(\frac{5}{four}\)
Answer:

Question 35.
focus: (0, \(\frac{6}{vii}\))
vertex: (0, 0)
Answer:

Question 36.
focus: (-\(\frac{iv}{five}\), 0)
vertex: (0, 0)
Reply:

In Exercises 37–40, write an equation of the parabola shown.

Question 37.

Answer:

Question 38.

Answer:

Question 39.

Reply:

Question 40.

Answer:

In Exercises 41–46, identify the vertex, focus, directrix, and centrality of symmetry of the parabola. Describe the transformations of the graph of the standard equation with p = 1 and vertex (0, 0).

Question 41.
y = \(\frac{1}{8}\)(ten – 3)² + ii
Answer:

Question 42.
y = –\(\frac{1}{four}\)(ten + 2)² + 1
Answer:

Question 43.
10 = \(\frac{1}{16}\)(y – three)² + 1
Answer:

Question 44.
y = (x + 3)² – 5
Answer:

Question 45.
x = -3(y + 4)² + 2
Reply:

Question 46.
x = four(y + five)² – 1
Answer:

Question 47.
MODELING WITH MATHEMATICS Scientists studying dolphin echolocation simulate the projection of a bottlenose dolphin'southward clicking sounds using calculator models. The models originate the sounds at the focus of a parabolic reflector. The parabola in the graph shows the cross section of the reflector with focal length of i.3 inches and discontinuity width of 8 inches. Write an equation to represent the cross section of the reflector. What is the depth of the reflector?

Answer:

Question 48.
MODELING WITH MATHEMATICS Solar free energy tin exist concentrated using long troughs that have a parabolic cantankerous section equally shown in the figure. Write an equation to represent the cross section of the trough. What are the domain and range in this situation? What do they represent?

Answer:

Question 49.
Abstruse REASONING As | p | increases, how does the width of the graph of the equation y = \(\frac{1}{iv p}\)x² change? Explain your reasoning.
Answer:

Question 50.
HOW DO YOU Meet IT? The graph shows the path of a volleyball served from an initial superlative of half dozen feet as it travels over a net.

a. Label the vertex, focus, and a point on the directrix.
b. An underhand serve follows the same parabolic path merely is hit from a height of 3 feet. How does this touch on the focus? the directrix?
Answer:

Question 51.
CRITICAL THINKING The altitude from point P to the directrix is 2 units. Write an equation of the parabola.

Reply:

Question 52.
Thought PROVOKING Ii parabolas have the aforementioned focus (a, b) and focal length of two units. Write an equation of each parabola. Place the directrix of each parabola.
Respond:

Question 53.
REPEATED REASONING Use the Altitude Formula to derive the equation of a parabola that opens to the correct with vertex (0, 0), focus (p, 0), and directrix x = -p.

Reply:

Question 54.
Trouble SOLVING The latus rectum of a parabola is the line segment that is parallel to the directrix, passes through the focus, and has endpoints that lie on the parabola. Notice the length of the latus rectum of the parabola shown.

Respond:

Maintaining Mathematical Proficiency

Write an equation of the line that passes through the points.(Section 1.3)

Question 55.
(1, -4), (2, -1)
Answer:

Question 56.
(-3, 12), (0, half-dozen)
Answer:

Question 57.
(three, one), (-five, 5)
Answer:

Question 58.
(2, -1), (0, 1)
Answer:

Utilize a graphing calculator to find an equation for the line of best fit.

Question 59.

Answer:

Question 60.

Answer:

Lesson 2.4 Modeling with Quadratic Functions

Essential Question
How can you use a quadratic office to model a real-life situation?

EXPLORATION ane
Modeling with a Quadratic Function

Work with a partner. The graph shows a quadratic office of the form
P(t) = at² + bt + c

which approximates the yearly profits for a company, where P(t) is the profit in year t.
a. Is the value of a positive, negative, or nix? Explain.
b. Write an expression in terms of a and b that represents the twelvemonth t when the company made the least profit.
c. The company fabricated the same yearly profits in 2004 and 2012. Estimate the year in which the company fabricated the least profit.
d. Assume that the model is still valid today. Are the yearly profits currently increasing, decreasing, or abiding? Explicate.

EXPLORATION ii
Modeling with a Graphing Reckoner
Work with a partner. The table shows the heights h (in feet) of a wrench t seconds after it has been dropped from a building under construction.

a. Use a graphing reckoner to create a besprinkle plot of the data, as shown at the right. Explicate why the data appear to fit a quadratic model.
b. Utilize the quadratic regression feature to detect a quadratic model for the information.
c. Graph the quadratic function on the same screen as the scatter plot to verify that it fits the data.


d. When does the wrench hit the footing? Explicate.

Communicate Your Answer

Question 3.
How tin can you apply a quadratic function to model a real-life situation?

Question 4.
Utilise the Internet or some other reference to find examples of existent-life situations that can exist modeled by quadratic functions.

ii.4 Lesson

Monitoring Progress

Question 1.
WHAT IF? The vertex of the parabola is (50, 37.v). What is the summit of the cyberspace?

Question two.
Write an equation of the parabola that passes through the point (-1, 2) and has vertex (4, -9).

Question 3.
WHAT IF? The y-intercept is 4.viii. How does this change your answers in parts (a) and (b)?

Question 4.
Write an equation of the parabola that passes through the point (2, v) and has x-intercepts -ii and 4.

Question v.
Write an equation of the parabola that passes through the points (-one, 4), (0, i), and (two, 7).

Question 6.
The table shows the estimated profits y (in dollars) for a concert when the charge is 10 dollars per ticket. Write and evaluate a office to determine what the accuse per ticket should be to maximize the profit.

Question 7.
The table shows the results of an experiment testing the maximum weights y (in tons) supported by water ice ten inches thick. Write a function that models the data. How much weight can exist supported past ice that is 22 inches thick?

Modeling with Quadratic Functions 2.4 Exercises

Vocabulary and Core Concept Cheque

Question 1.
WRITING Explain when information technology is advisable to employ a quadratic model for a set of data.
Answer:

Question 2.
DIFFERENT WORDS, Aforementioned QUESTION
Which is different? Detect "both" answers.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–eight, write an equation of the parabola in vertex form.

Question 3.

Answer:

Question iv.

Answer:

Question 5.
passes through (13, 8) and has vertex (three, ii)
Answer:

Question half dozen.
passes through (-7, -15) and has vertex (-five, 9)
Answer:

Question vii.
passes through (0, -24) and has vertex (-half dozen, -12)
Answer:

Question viii.
passes through (6, 35) and has vertex (-one, 14)
Respond:

In Exercises 9–14, write an equation of the parabola in intercept form.

Question 9.

Answer:

Question 10.

Answer:

Question 11.
x-intercepts of 12 and -6; passes through (fourteen, 4)
Answer:

Question 12.
x-intercepts of ix and one; passes through (0, -18)
Answer:

Question 13.
x-intercepts of -sixteen and -two; passes through (-eighteen, 72)
Answer:

Question 14.
x-intercepts of -7 and -iii; passes through (-2, 0.05)
Reply:

Question xv.
WRITING Explain when to use intercept grade and when to use vertex form when writing an equation of a parabola.
Answer:

Question xvi.
ANALYZING EQUATIONS Which of the following equations represent the parabola?

A. y = two(10 – two)(x + one)
B. y = 2(x + 0.5)^two – 4.5
C. y = 2(x – 0.5)² – 4.5
D. y = 2(x + 2)(10 – one)
Respond:

In Exercises 17–xx, write an equation of the parabola in vertex form or intercept form.

Question 17.

Reply:

Question 18.

Answer:

Question xix.

Reply:

Question twenty.

Answer:

Question 21.
ERROR ANALYSIS Describe and correct the mistake in writing an equation of the parabola.

Answer:

Question 22.
MATHEMATICAL CONNECTIONS The area of a rectangle is modeled by the graph where y is the area (in foursquare meters) and ten is the width (in meters). Write an equation of the parabola. Find the dimensions and respective area of one possible rectangle. What dimensions outcome in the maximum surface area?

Answer:

Question 23.
MODELING WITH MATHEMATICS Every rope has a rubber working load. A rope should not be used to lift a weight greater than its prophylactic working load. The table shows the safe working loads S (in pounds) for ropes with circumference C (in inches). Write an equation for the safe working load for a rope. Find the safe working load for a rope that has a circumference of x inches.

Reply:

Question 24.
MODELING WITH MATHEMATICS A baseball is thrown up in the air. The table shows the heights y (in feet) of the baseball game subsequently x seconds. Write an equation for the path of the baseball. Find the height of the baseball after 1.7 seconds.

Respond:

Question 25.
COMPARING METHODS You utilize a organization with three variables to find the equation of a parabola that passes through the points (−8, 0), (ii, −20), and (1, 0). Your friend uses intercept form to find the equation. Whose method is easier? Justify your respond.
Answer:

Question 26.
MODELING WITH MATHEMATICS The table shows the distances y a motorcyclist is from home afterwards ten hours.

a. Determine what type of role you lot can employ to model the data. Explain your reasoning.
b. Write and evaluate a role to make up one's mind the distance the motorcyclist is from home afterwards 6 hours.
Respond:

Question 27.
USING TOOLS The table shows the heights h (in anxiety) of a sponge t seconds later it was dropped by a window cleaner on top of a skyscraper.

a. Use a graphing estimator to create a scatter plot. Which better represents the data, a line or a parabola? Explain.
b. Employ the regression feature of your reckoner to find the model that best fits the data.
c. Use the model in function (b) to predict when the sponge volition hit the ground.
d. Identify and interpret the domain and range in this situation.
Answer:

Question 28.
MAKING AN Argument Your friend states that quadratic functions with the aforementioned 10-intercepts have the same equations, vertex, and axis of symmetry. Is your friend correct? Explain your reasoning.
Answer:

In Exercises 29–32, clarify the differences in the outputs to determine whether the data are linear, quadratic, or neither. Explicate. If linear or quadratic, write an equation that fits the data.

Question 29.

Answer:

Question 30.

Answer:

Question 31.

Answer:

Question 32.

Answer:

Question 33.
PROBLEM SOLVING The graph shows the number y of students absent from school due to the influenza each 24-hour interval ten.

a. Interpret the significant of the vertex in this situation.
b. Write an equation for the parabola to predict the number of students absent on solar day 10.
c. Compare the boilerplate rates of change in the students with the flu from 0 to half-dozen days and 6 to 11 days.
Answer:

Question 34.
Idea PROVOKING Describe a real-life situation that can be modeled by a quadratic equation. Justify your respond.
Answer:

Question 35.
Problem SOLVING The table shows the heights y of a competitive water-skier 10 seconds later jumping off a ramp. Write a role that models the height of the water-skier over time. When is the h2o-skier five feet above the water? How long is the skier in the air?

Reply:

Question 36.
HOW DO YOU Run across IT? Employ the graph to make up one's mind whether the average rate of modify over each interval is positive, negative, or nil.

a. 0 ≤ x ≤ 2
b. two ≤ x ≤ five
c. ii ≤ x ≤ 4
d. 0 ≤ x ≤ four
Answer:

Question 37.
REPEATED REASONING The tabular array shows the number of tiles in each figure. Verify that the data show a quadratic relationship. Predict the number of tiles in the 12th effigy.

Respond:

Maintaining Mathematical Proficiency

Cistron the trinomial. (Skills Review Handbook)

Question 38.
x² + 4x + 3
Answer:

Question 39.
10ⁱⁱ – 3x + 2
Answer:

Question 40.
3x^two – 15x + 12
Answer:

Question 41.
5x² + 5x – 30
Answer:

Quadratic Functions Performance Task: Accident Reconstruction

2.3–2.4 What Did You Learn?

Core Vocabulary
focus, p. 68
directrix, p. 68

Cadre Concepts

Section two.3
Standard Equations of a Parabola with Vertex at the Origin, p. 69
Standard Equations of a Parabola with Vertex at (h, 1000), p. 70

Section 2.4
Writing Quadratic Equations, p. 76
Writing Quadratic Equations to Model Data, p. 78

Mathematical Practices

Question 1.
Explain the solution pathway yous used to solve Exercise 47 on page 73.

Question two.
Explicate how y'all used definitions to derive the equation in Practise 53 on folio 74.

Question 3.
Explain the shortcut y'all institute to write the equation in Practise 25 on page 81.

Question 4.
Describe how you were able to construct a viable argument in Exercise 28 on page 81.

Performance Job

Accident Reconstruction

Was the commuter of a machine speeding when the brakes were applied? What do skid marks at the scene of an accident reveal well-nigh the moments before the standoff?

To explore the answers to these questions and more, go to BigIdeasMath.com.

Quadratic Functions Affiliate Review

Draw the transformation of f(ten) = x² represented by yard. Then graph each role.

Question ane.
g(x) = (x + iv)²

Question 2.
grand(x) = (10 – 7)^{2 + 2}

Question 3.
grand(x) = -3(x + 2)² – 1

Question 4.
Let the graph of k be a horizontal shrink by a factor of \(\frac{2}{3}\), followed by a translation 5 units left and 2 units downward of the graph of f(x) = x².

Question 5.
Let the graph of g be a translation ii units left and three units up, followed by a reflection in the y-centrality of the graph of f(x) = ten² – 2x.

Graph the function. Label the vertex and centrality of symmetry. Find the minimum or maximum value of f. Describe where the part is increasing and decreasing.

Question 6.
f(ten) = iii(10 – 1)² – 4

Question 7.
g(x) = -2x² + 16x + iii

Question 8.
h(ten) = (x – 3)(x + 7)

Question 9.
Yous tin make a solar hot-canis familiaris cooker by shaping foil-lined cardboard into a parabolic trough and passing a wire through the focus of each end slice. For the trough shown, how far from the bottom should the wire be placed?

Question 10.
Graph the equation 36y = x². Identify the focus, directrix, and axis of symmetry.

Write an equation of the parabola with the given characteristics.

Question 11.
vertex: (0, 0)
directrix: x = ii

Question 12.
focus: (ii, 2)
vertex: (ii, half-dozen)

Write an equation for the parabola with the given characteristics.

Question thirteen.
passes through (ane, 12) and has vertex (10, -iv)

Question 14.
passes through (4, 3) and has x-intercepts of -1 and 5

Question fifteen.
passes through (-2, 7), (1, 10), and (2, 27)

Question xvi.
The table shows the heights y of a dropped object subsequently x seconds. Verify that the data testify a quadratic human relationship. Write a function that models the information. How long is the object in the air?

Quadratic Functions Chapter Test

Question i.
A parabola has an centrality of symmetry y= 3 and passes through the point (2, 1). Discover another betoken that lies on the graph of the parabola. Explain your reasoning.

Question 2.
Let the graph of g be a translation ii units left and 1 unit downward, followed past a reflection in the y-axis of the graph of f(x) = (2x + i)² – iv. Write a dominion for g.

Question 3.
Identify the focus, directrix, and centrality of symmetry of x = 2yⁱⁱ. Graph the equation.

Question 4.
Explain why a quadratic function models the information. Then use a linear organisation to find the model.

Write an equation of the parabola. Justify your respond.

Question 5.

Question half-dozen.

Question 7.

Question viii.
A surfboard shop sells xl surfboards per month when it charges $500 per surfboard. Each time the shop decreases the toll by $10, it sells one boosted surfboard per calendar month. How much should the shop charge per surfboard to maximize the corporeality of money earned? What is the maximum amount the store tin can earn per month? Explain.

Question 9.
Graph f(x) = 8x² – 4x+ 3. Characterization the vertex and centrality of symmetry. Depict where the function is increasing and decreasing.

Question 10.
Sunfire is a machine with a parabolic cross department used to collect solar energy. The Dominicus's rays are reflected from the mirrors toward ii boilers located at the focus of the parabola. The boilers produce steam that powers an alternator to produce electricity.
a. Write an equation that represents the cross department of the dish shown with its vertex at (0, 0).
b. What is the depth of Sunfire? Justify your answer.

Question 11.
In 2011, the price of gilded reached an all-time high. The table shows the prices (in dollars per troy ounce) of gold each year since 2006 (t = 0 represents 2006). Detect a quadratic part that all-time models the data. Employ the model to predict the price of gilt in the twelvemonth 2016.

Quadratic Functions Cumulative Assessment

Question 1.
You and your friend are throwing a football. The parabola shows the path of your friend'south throw, where ten is the horizontal distance (in feet) and y is the corresponding tiptop (in feet). The path of your throw can be modeled by h(x) = −16x^two + 65x + 5. Choose the correct inequality symbol to indicate whose throw travels higher. Explicate your reasoning.

Question two.
The role chiliad(x) = \(\frac{one}{ii}\)∣x − 4 ∣ + 4 is a combination of transformations of f(10) = | x|. Which combinations describe the transformation from the graph of f to the graph of thousand?
A. translation 4 units right and vertical shrink by a cistron of \(\frac{one}{2}\), followed by a translation four units upwardly
B. translation iv units right and 4 units up, followed past a vertical shrink by a factor of \(\frac{1}{2}\)
C. vertical shrink past a factor of \(\frac{1}{2}\) , followed by a translation four units up and 4 units right
D. translation 4 units right and 8 units up, followed past a vertical compress by a gene of \(\frac{1}{2}\)

Question three.
Your school decides to sell tickets to a dance in the school cafeteria to raise money. In that location is no fee to utilize the cafeteria, merely the DJ charges a fee of $750. The table shows the profits (in dollars) when x students nourish the dance.

a. What is the cost of a ticket?
b. Your school expects 400 students to nourish and finds another DJ who only charges $650. How much should your school charge per ticket to withal brand the same profit?
c. Your school decides to charge the amount in office (a) and use the less expensive DJ. How much more than money volition the school raise?

Question 4.
Order the following parabolas from widest to narrowest.
A. focus: (0, −3); directrix: y = 3
B. y = \(\frac{i}{xvi}\)x² + 4
C. x = \(\frac{1}{8}\)y²
D. y = \(\frac{i}{4}\)(x − ii)ⁱⁱ + 3

Question 5.
Your friend claims that for g(x) = b, where b is a real number, there is a transformation in the graph that is impossible to notice. Is your friend correct? Explicate your reasoning.

Question 6.
Let the graph of g stand for a vertical stretch and a reflection in the x-axis, followed by a translation left and down of the graph of f(x) = x^two. Utilize the tiles to write a rule for g.

Question vii.
Two balls are thrown in the air. The path of the start ball is represented in the graph. The second ball is released one.five feet higher than the first ball and after three seconds reaches its maximum superlative 5 anxiety lower than the first ball.

a. Write an equation for the path of the second brawl.
b. Do the assurance hitting the basis at the same fourth dimension? If so, how long are the assurance in the air? If not, which ball hits the footing kickoff? Explain your reasoning.

Question viii.
Let the graph of g be a translation iii units right of the graph of f. The points (−1, 6), (3, fourteen), and (6, 41) lie on the graph of f. Which points lie on the graph of g?
A. (2, half-dozen)
B. (2, 11)
C. (vi, fourteen)
D. (half dozen, 19)
Due east. (9, 41)
F. (ix, 46)

Question 9.
Gym A charges $ten per month plus an initiation fee of $100. Gym B charges $thirty per calendar month, simply due to a special promotion, is not currently charging an initiation fee.
a. Write an equation for each gym modeling the full toll y for a membership lasting x months.
b. When is it more than economical for a person to choose Gym A over Gym B?
c. Gym A lowers its initiation fee to $25. Describe the transformation this modify represents and how it affects your decision in part (b).

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