Mathematics 9 Quarter 1 Module 19: Characteristics of the Graphs of Quadratic Function

Mathematics Quarter 1 – Module 19 Characteristics of the Graphs of Quadratic Function

Week 7 Learning Code – M9AL-Ih-13.2

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Quarter 1 – Module 19 – New Normal Math for G9 First Edition 2020 Copyright © 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writers: Maricon G. Manio – MTI Analynn M. Argel – MTII

Editor: Sally C. Caleja – Head Teacher VI Maita G. Camilon – Head Teacher VI Editha T. Moredo – Head Teacher VI

Validators: Remylinda T. Soriano, EPS, Math Angelita Z. Modesto, PSDS George B. Borromeo, PSDS

Illustrator: Writers

Layout Artist: Writers

Management Team: Malcolm S. Garma, Regional Director Genia V. Santos, CLMD Chief Dennis M. Mendoza, Regional EPS in Charge of LRMS and Regional ADM Coordinator Maria Magdalena M. Lim, CESO V, Schools Division Superintendent Aida H. Rondilla, Chief-CID Lucky S. Carpio, Division EPS in Charge of LRMS and Division ADM Coordinator

MODULE 19

The previous module, you learned to graph quadratic functions. You were given opportunities to explore the basic characteristics of parabola. In this module, you will find out all the characteristics of the graphs of quadratic functions.

What I Need to Know

LEARNING COMPETENCY

The learners will be able to:

• determine the following (a) domain (b) range (c) intercepts (d) axis of symmetry (e) vertex (f) direction of the opening of the parabola from the given graph of the equation. M9AL-Ih-13.2

What I Know

Find out how much you already know about the characteristics of the graphs of quadratic functions. Write the letter that you think is the correct answer to each question on your answer sheet. Answer all items. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.

1. What is the graph of a quadratic function?
   A) Circle B) Line C) Parabola D) V-Shape

   
   

2. It is the turning point of the parabola, which indicates the highest/lowest point of the graph.
   A) Axis of symmetry B) Range C) Vertex D) Zeros

   
   

3. What do you call the x-coordinates of the points of the parabola that lie on the x-axis?
   A) Axis of symmetry B) Range C) Vertex D) Zeros

   
   

4. Which of the following is the vertex form of the quadratic function y = ax² + bx + c with vertex at (h, k)?
   A) y = k(x − h)² + a B) y = h(x − a)² + k
   C) y = a(x − k)² + h D) y = a(x − h)² + k

   
   

5. What is the vertex form of the equation y = x² + 4x + 7?
   A) y = (x − 2)² − 3 B) y = (x − 2)² + 3
   C) y = (x + 2)² − 3 D) y = (x + 2)² + 3

   
   

6. What is the vertex of the parabola having the equation y = (x – 1)² + 3?
   A) (3, –1) B) (–1, 3) C) (1, 3) D) (3, 1)

   
   

7. What is the maximum value of the graph of the quadratic function y = −x² + 4x?
   A) – 4 B) – 2 C) 2 D) 4

   
   

8. What is the y-intercept of the parabola defined by the equation y = (x + 2)²?
   A) (0, – 4) B) (0, – 2) C) (0, 2) D) (0, 4)

   
   

9. Describe the graph of f(x) = x² + 6x – 24.
   A) Opens to the right B) Opens to the left
   C) Opens upward D) Opens downward

   
   

For items 10-14, refer to the graph at the right.

[chart: graph of a parabola opening downward with vertex near (0, 5), for items 10-14; no title printed; x-axis labels -5, -4, -3, -2, -1, 1, 2, 3, 4, x; y-axis labels -2, -1, 1, 2, 3, 4, 5, 6, y; origin O]

10. Which of the following could be the equation of the parabola?
    A) y = −x² + 5
    B) y = x²/2 + 5
    C) y = −x²/2 + 5
    D) y = x² + 5

    
    

11. What is the vertex of the parabola?
    A) (−5, 0) B) (0, −5) C) (0, 5) D) (5, 0)

    
    

12. What is the axis of symmetry of the graph?
    A) x = 0 B) y = 0 C) x = 5 D) y = 5

    
    

13. What is the range of the parabola?
    A) (−∞, −5] B) (−∞, +∞) C) (−∞, 5] D) [5, +∞)

    
    

14. Determine the y-intercept of the graph?
    A) (0, −5) B) (0, 5) C) (0, −3) D) (0, 3)

    
    

15. Which of the following describes the parabola having the equation y = x² + 3?
    A) It lies entirely above the x-axis.
    B) Its vertex lies on the axis.
    C) It lies entirely below the axis.
    D) It intersects the x-axis at two distinct points.

    
    

Lesson

[transcriber’s note: no lesson heading is printed on these pages; the sections below are the lesson sections in the order printed]

WHAT’S IN

Communication, and Critical Thinking

Let’s start by a review of graphing quadratic function.

Transform each quadratic function in the form of y = a (x – h)² + k, then graph using vertex and axis of symmetry.

1. y = −(x² − 4x) + 1
2. 2x² + 6x = 3 − y
3. 4x² − y = 12x − 7
4. 2 − x² = y − 3x

WHAT’S NEW

Communication, and Collaboration

Kick that Ball!

Mara kicked a ball into the air. The height y in feet of the ball after x seconds can be modeled by the quadratic function y = -16x² + 32x.

The arc formed by the ball is a curve called parabola.

Can you graph actual parabola formed by the given function above and answer the questions below?

1. What is the opening of the parabola?
2. What are the coordinates of the turning point of the parabola?
3. What is the equation of the line that divides the parabola into two equal parts?
4. At what point/s does the parabola intersect/s the x-axis?
5. At what point/s does the parabola intersect/s the y-axis?
6. For what values of x satisfy the given function based on the graph?
7. For what values of y satisfy the given function based on the graph?

How did you find the activity? Were you able to answer all the questions based on the graph? If not, let us investigate further and

[illustration: a person kicking a soccer ball, with the ball’s curved arc drawn in the air]

WHAT IS IT

Communication, Critical Thinking, and Collaboration

Let’s investigate!

Shown on the right is the graph of the quadratic function defined by

y = -16x² + 32x

[chart: graph of the parabola y = -16x² + 32x; no title printed; x-axis labels -2, -1, 1, 2, 3, 4, x; y-axis labels 2, 4, 6, 8, 10, 12, 14, 16, y; origin O; the vertex is marked at the top and the curve crosses the x-axis at 0 and 2]

Notice that the parabola opens downward. In some cases parabola opens upward. In the quadratic function y = ax² + bx + c, if the value of a > 0, then parabola opens upward and has a minimum point. If a < 0 the parabola opens downward and has a maximum point.

The turning point of a parabola is called its vertex with coordinates at (h, k). In this case, the vertex is at (1, 16).

There is a line called the axis of symmetry which divides the graph into two parts such that one half of the graph is a reflection of the other half. In this graph, the equation of the axis of symmetry is x = 1. If the quadratic function is expressed in the form of y = a(x − h)² + k the vertex is the point (h, k). The line x = h is the axis of symmetry and k is the minimum or maximum value of the function.

The graph intersect the x-axis at (0, 0) and (2,0). These are called x-intercepts. These are the points at which the graph of an equation crosses the x-axis. While, the graph intersects the y-axis at (0, 0). This is called the y-intercept. It is a point which the graph of an equation crosses the y-axis.

The domain of a quadratic function is the set of real numbers. The range depends on whether the parabola opens upward or downward. If it opens upward, then the range is the set {y/ y≥k}; if it opens downward then the range is the set {y/ y≤k}. In this graph, the range is {y /y ≤ 16}, expressed as Set Notation, or [16, +∞), expressed as Group Notation.

To fully understand the characteristics of the graph of a quadratic function, study the following examples.

Example 1. Determine the vertex, direction of the opening of the parabola, axis of symmetry, domain, range and intercepts given the graph.

Solution: Vertex: (1, -4) Opening of the parabola: Upward Vertex is a minimum point.

[chart: graph of a parabola opening upward for Example 1; no title printed; x-axis labels -4, -3, -2, -1, 1, 2, 3, 4, 5, x; y-axis labels -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, y (as printed on the axis); origin O; the curve crosses the x-axis at -1 and 3 and the y-axis at -3; the vertex is at (1, -4)]

Axis of symmetry: x = 1 Domain: { x/ x ∈ ℝ } Range: { y / y ≥ -4 } Intercepts: x – intercepts (-1, 0) and (3, 0) y – intercept (0, -3)

Example 2. Determine the vertex, direction of the opening of the parabola, axis of symmetry, domain, range and intercepts given the graph.

Solution: Vertex: (-1, 4) Opening of the parabola: Downward Vertex is a maximum point. Axis of symmetry: x = -1 Domain: { x/ x ∈ ℝ } Range: { y / y ≤ 4 } Intercepts: x – intercepts (1, 0) and (-3, 0) y – intercept (0, 3)

Notice how vertex plays an important role in determining the characteristics of the graph of a quadratic function. Thus, even without the graph and you know the vertex, you can identify the direction of the opening of the parabola, axis of symmetry, domain, range and intercepts given the graph. Study the next example.

Example 3. Determine the vertex, direction of the opening of the parabola, axis of symmetry, domain, range and intercepts of the quadratic function

y = −(x + 2)² − 1

Solution a. Vertex : (h, k) of the form y = a(x − h)² + k is (-2, -1) b. Direction of the opening of the parabola : since a < 0, then it opens downward c. Axis of symmetry : Since x = -2, then the equation of the axis of symmetry is x= 2 [transcriber’s note: the axis of symmetry is printed as “x= 2”; for the vertex (-2, -1) the axis is x = -2, as stated in the line above] d. Domain : the domain of quadratic function is all real numbers e. Range : since it opens downward then the range is {y: y ≤ k}, {y: y ≤ 7} or [7, -∞) f. Intercepts : x – intercepts, let y be zero then compute for the value of x The x-intercepts are at (3, 0) and (1, 0) y – intercept, let x be zero then compute for the value of y The y-intercept is at (0, 3).

Note: (if the equation in the form of y = ax² + bx + c, then y – intercept is at (0, c).

[chart: graph of a parabola opening downward for Example 2; no title printed; x-axis labels -4, -3, -2, -1, 1, 2, 3, 4, x; y-axis labels -4, -3, -2, -1, 1, 2, 3, 4, y; origin O; the curve crosses the x-axis at 1 and -3 and the y-axis at 3; the vertex is at (-1, 4)]

WHAT’S MORE

Critical Thinking, Communication and Collaboration

Draw and Describe My Graph!

Sketch the graph of each quadratic function and identify the vertex, direction of the opening of the parabola, domain, range and intercepts given the equation. State whether the vertex is a minimum or a maximum point and write the equation of the axis of symmetry. (Hint: Convert the quadratic function in standard form.)

1. y = (x + 1)² − 4

[chart: blank coordinate grid for item 1; x-axis labels -4, -3, -2, -1, 1, 2, 3, 4, x; y-axis labels -4, -3, -2, -1, 1, 2, 3, 4, y; origin O]

Vertex: _________________ Opening of the graph: __________ Vertex is a ______________ point. Axis of symmetry: _________ Domain: _______ Range : _______ x – intercept/s: ______________ y – intercept/s: ______________

2. y = −(2x² − 4x − 1)

[chart: blank coordinate grid for item 2; x-axis labels -4, -3, -2, -1, 1, 2, 3, 4, x; y-axis labels -4, -3, -2, -1, 1, 2, 3, 4, y; origin O]

Vertex: __________________ Opening of the graph: __________ Vertex is a ______________ point. Axis of symmetry: _________ Domain: _______ Range : _______ x – intercept/s : ______________ y – intercept/s : ______________

Vertex: _________________ Opening of the graph: __________ Vertex is a ______________ point. Axis of symmetry: _________ Domain: _______ Range : _______ x – intercept/s : ______________ y – intercept/s : ______________

3. 2x² + y = 8x − 6

[chart: blank coordinate grid for item 3; axes labeled x and y; x tick labels: -2, -1, 1, 2, 3, 4, 5, 6; y tick labels: 2, 1, -1, -2, -3, -4, -5, -6]

4. y − 6x − 1 = −3x²

Vertex: _________________ Opening of the graph: __________ Vertex is a ______________ point. Axis of symmetry: _________ Domain: _______ Range : _______ x – intercept/s : ______________ y – intercept/s : ______________

[chart: blank coordinate grid for item 4; axes labeled x and y; x tick labels: -2, -1, 1, 2, 3, 4, 5, 6; y tick labels: 4, 3, 2, 1, -1, -2, -3, -4]

How did you find the activity? Were you able to graph all the quadratic functions? Were you able to identify the characteristics of the parabola? If not, in which part did you find challenging? How did you cope up with it?

From the graph of the following quadratic functions in module 18, determine their vertex, direction of the opening of the parabola, axis of symmetry, domain, range and y – intercept. (Hint: Convert the quadratic function in standard form.)

1. y = (x + 2)² – 3 6. y = -3x² – 6x + 4
2. y = -(x – 3)² + 2 7. y = 2x² – 12x + 9
3. y = 2x² – 8x + 3 8. 2x² – 16x = y – 33
4. y = -x² + 2x + 1 9. y – 3x² +6x = 5
5. y + 3x² = 6x – 1 10. y – 4 = 3x² + 12x

WHAT I HAVE LEARNED

Characteristics of the graph of quadratic function

The graph of a quadratic is a smooth curve called parabola.

• Vertex – the minimum or the maximum point on the quadratic function. If the quadratic function is expressed in the form of y = a(x − h)² + k the vertex is the point (h, k). If it is in the form of y = ax² + bx + c use the formula h = −b/2a and k = f(h).
• Domain – The domain of a quadratic function is the set of real numbers
• Range – The range depends on whether the parabola opens upward or downward. If it opens upward, then the range is the set {y: y ≥ k}; if it opens downward then the range is the set {y: y ≤ k}.
• Axis of symmetry – divides the graph into two parts such that one half of the graph is a reflection of the other half. The equation of the line is x = h.
• Direction of the opening of the parabola – If the value of > 0, then parabola opens upward and has a minimum point. If < 0, then parabola opens downward and has a maximum point. [transcriber’s note: the variable after “value of” does not appear in the rendered page; the symbol is missing]
• Intercepts – x – intercept (let y be zero then solve for x) y – intercept (y = c)

WHAT I CAN DO

Critical Thinking

ASSESSMENT

Read each item carefully. Identify the choice that correctly completes the statement or answers the question

1. Which of the following describes the parabola having the equation y = −x² − 3?
   • A) It lies entirely above the x-axis.
   • B) Its vertex lies on the axis.
   • C) It lies entirely below the axis.
   • D) It intersects the x-axis at two distinct points.

For items 2-5, refer to the graph at the right.

[chart: parabola graph referred to in items 2–5; x tick labels: -2, -1, 1, 2, 3, 4, 5, 6; y tick labels: 1 to 8]

2. Which of the following could be the equation of the parabola?

   • A) y = (x − 2)² + 1
   • B) y = (x + 1)² + 2
   • C) y = (x + 2)² + 1
   • D) y = (x − 1)² + 2

3. Determine the y- intercept of the graph?

   • A) (0, −5)
   • B) (0, 5)
   • C) (0, −3)
   • D) (0, 3)

4. What is the range of the parabola?

   • A) (−∞, −1]
   • B) (−∞, +∞)
   • C) (−∞, 1]
   • D) [1, +∞)

5. What is the axis of symmetry of the graph?

   • A) x = 2
   • B) y = 2
   • C) x = 1
   • D) y = 1

6. What is the vertex of the parabola?

   • A) (−2, −1)
   • B) (2, 1)
   • C) (−2, 1)
   • D) (2, −1)

7. Describe the graph of f(x) = −x² + 6x − 24.

   • A) Opens to the right
   • B) Opens to the left
   • C) Opens upward
   • D) Opens downward

8. What is the y-intercept of the parabola defined by the equation y = (x + 2)² ?

   • A) (0, −4)
   • B) (0, −2)
   • C) (0, 2)
   • D) (0, 4)

9. What is the minimum value of the graph of the quadratic function y = x² + 4x?

   • A) −4
   • B) −2
   • C) 2
   • D) 4

10. What is the vertex of the parabola having the equation y = (x − 3)² + 1 ?

    • A) (3, −1)
    • B) (−1, 3)
    • C) (1, 3)
    • D) (3, 1)

ADDITIONAL ACTIVITIES

Communication, Critical Thinking, Creativity and Character Building

The General Projectile Function

If an object is thrown straight up from the top of a building h feet tall with an initial velocity of v feet per second, then the height of the object as a function of time can be modeled by the quadratic function h(t) = –16t² + vt + h, where h(t) is the height of the object (in feet) t seconds after it is thrown. This is the general projectile function.

If Mara hit the ball at 3 ft. above the ground with an initial velocity of 32 ft/sec., the path of the ball is given by the function h(t) = −16t² + 32t + 3.

If Clara kicked a ball at 4ft above the ground with an initial level of velocity of 16ft/sec, then the path of the ball is given by the function h(t) = −16t² + 16t + 4.

Your Task!

Without graphing determine the vertex, direction of the opening of the parabola, axis of symmetry, domain, range and y – intercept of each of the general projectile function given above.

PROBLEM – BASED WORKSHEET

The Frame Designer

Jeanna is assigned to design a rectangular mini frame of perimeter 12 inches.

Let’s Analyze!

1. Regardless of dimensions as long as the perimeter is 12, complete the table for the width, length and Area.

width (w)	1	2	3	4	5	w
length
Area

2. Write a function A(w) for the area of the frame in terms of the width.

   
   

3. Graph the function A(w). Label each axis.

   
   

4. Determine the following:

   • a. Domain
   • b. Range
   • c. Vertex
   • d. Axis of symmetry
   • e. Direction of the opening of the parabola

5. Determine the maximum area of the frame that Jeanna can design. What are the dimensions of this screen?

   
   

E-Search

You may also check the following link for your reference and further learnings on the characteristics of the graphs of quadratic function.

• https://www.khanacademy.org/math/algebra-home/alg-functions/alg-determining-the-range-of-a-function/v/domain-and-range-of-a-function-given-a-formula
• https://www.youtube.com/watch?v=V2udel2WylU
• https://www.youtube.com/watch?v=VdVPg04t_6w

REFERENCES

http://www.mesacc.edu/~scotz47781/mat120/notes/projectile_motion/projectile_motion.html#:~:text=Projectile%20Motion,seconds%20after%20it%20is%20thrown. https://www.slideshare.net/paolodagaojes/9-math-lm-u1m2v10 https://www.slideshare.net/dionesioable/module-2-quadratic-functions https://sciencing.com/everyday-examples-situations-apply-quadratic-equations-10200.html

Illustrations: https://www.freepik.com/free-vector/woman-with-long-hair-teaching-online_7707557.htm https://www.freepik.com/free-vector/kids-having-online-lessons_7560046.htm

[transcriber’s note: this page is printed upside down in the source; the text below is transcribed in reading order after rotating it]

ADDITIONAL ACTIVITIES

For h(t) = −16t² + 32t + 3 or h(t) = −16(t − 1)² + 19

Vertex: (1, 19) Opening of the graph: Downward Axis of symmetry: x = 1 Domain: {x/ x ∈ ℝ } Range: {y/ y ≤ 19} y – intercept: (0, 3)

For h(t) = −16t² + 16t + 4 or h(t) = −16(t − 1/2)² + 8

Vertex: (1/2, 8) Opening of the graph: Downward Axis of symmetry: x = 1/2 Domain: {x/ x ∈ ℝ } Range: {y/ y ≤ 8} y – intercept: (0, 4)

PROBLEM – BASED WORKSHEET

width (w)	1	2	3	4	5	w
length	5	4	3	2	1	6 − w
Area	5	8	9	8	5	w(6 − w)

2. A(w) = −w² − 6w [?] [transcriber’s note: the sign of the terms is hard to read in the rotated image]

   
   

3. Graph the function A(w). Label each axis.

   
   

[chart: parabola graph of A(w), drawn opening downward with its vertex near the top of the grid in document orientation; axis tick labels are not legible in the rotated image]

4. Determine the following:

   • a. Domain: x ∈ ℝ
   • b. Range: y ≤ 9
   • c. Vertex (3, 9)
   • d. Axis of symmetry: x = 3
   • e. Direction of the opening of the parabola: downward

5. The maximum area of the frame that Jeanna can design is 9 square inches with dimensions 3 inches by 3 inches.

   
   

[transcriber’s note: this page is printed upside down in the source; the text below is transcribed in reading order after rotating it]

5. y = −3(x + 3)² + 26

Vertex: (−3, 26) Opening of the graph: Downward Axis of symmetry: x = −3 Domain: {x/ x ∈ ℝ } Range: {y/ y ≤ 26} y – intercept: (0, −1)

6. y = −3(x + 1)² + 7

Vertex: (−1, 7) Opening of the graph: Downward Axis of symmetry: x = −1 Domain: {x/ x ∈ ℝ } Range: {y/ y ≤ 7} y – intercept: (0, 4)

7. y = 2(x − 3)² + 27

Vertex: (3, 27) Opening of the graph: Upward Axis of symmetry: x = 3 Domain: {x/ x ∈ ℝ } Range: {y/ y ≥ 27} y – intercept: (0, 9)

8. y = 2(x + 4)² + 1 [?]

Vertex: (−4, 1) Opening of the graph: Upward Axis of symmetry: x = −4 Domain: {x/ x ∈ ℝ } Range: {y/ y ≥ 1} y – intercept: (0, 33)

[transcriber’s note: the sign inside the parentheses of item 8 is hard to read in the rotated image; the vertex and axis as printed correspond to (x + 4)]

9. y = 3(x − 1)² + 2

Vertex: (1, 2) Opening of the graph: Upward Axis of symmetry: x = 1 Domain: {x/ x ∈ ℝ } Range: {y/ y ≥ 2} y – intercept: (0, 5)

10. y = 3(x + 2)² − 8

Vertex: (−2, −8) Opening of the graph: Upward Axis of symmetry: x = −2 Domain: {x/ x ∈ ℝ } Range: {y/ y ≥ −8} y – intercept: (0, 4)

ASSESSMENT

1. C 3. A 5. D 7. C 9. D
2. D 4. B 6. A 8. D 10. C

[transcriber’s note: this page is printed upside down in the source; the text below is transcribed in reading order after rotating it]

4. y − 6x − 1 = −3x² or y = −3(x − 1)² + 4

Vertex: (1, 4) Opening of the graph: Downward Vertex is a maximum point Axis of symmetry: x = 1 Domain: {x/ x ∈ ℝ } Range: {y/ y ≤ 4}

[chart: parabola graph for item 4 of the What I Can Do answers; axis tick labels are not legible in the rotated image]

WHAT I CAN DO

1. y = (x + 2)² − 3

Vertex: (−2, −3) Opening of the graph: Upward Axis of symmetry: x = −2 Domain: {x/ x ∈ ℝ } Range: {y/ y ≥ −3} y – intercept: (0, 1)

2. y = −(x − 3)² + 2

Vertex: (3, 2) Opening of the graph: Downward Axis of symmetry: x = 3 Domain: {x/ x ∈ ℝ } Range: {y/ y ≤ 2} y – intercept: (0, −7)

3. y = (x − 2)² − 5

Vertex: (2, −5) Opening of the graph: Upward Axis of symmetry: x = 2 Domain: {x/ x ∈ ℝ } Range: {y/ y ≥ −5} y – intercept: (0, −1)

What’s New

1. downward
2. (1, 16)
3. x = 1
4. (0,0) and (2,1)
5. (0, 0)
6. set of Real numbers or {x/ x ∈ ℝ}
7. {y/y≤ 16}

What’s More

1. y = (x + 1)² − 4

[chart: parabola graph on a coordinate grid; no number or title printed]

Vertex: (-1, -4) Opening of the graph: Upward Vertex is a minimum point Axis of symmetry: x = -1 Domain: {x/ x ∈ ℝ} Range: {y/ y ≥ -4}

2. y = -(2x² − 4x − 1) or y = −2(x − 1)² + 3

[chart: parabola graph on a coordinate grid; no number or title printed]

Vertex: (1, 3) Opening of the graph: Downward Vertex is a maximum point Axis of symmetry: x = 2 Domain: {x/ x ∈ ℝ} Range: {y/ y ≤ 3}

3. y + 2x² = 8x − 6 or y = −2(x − 2)² + 2

[chart: parabola graph on a coordinate grid; no number or title printed]

Vertex: (2, 2) Opening of the graph: Downward Vertex is a maximum point Axis of symmetry: x = 2 Domain: {x/ x ∈ ℝ} Range: {y/ y ≤ 2}

Answer Key

What I Know

1. C | 6. C | 11. C
2. C | 7. D | 12. A
3. D | 8. D | 13. C
4. D | 9. C | 14. B
5. D | 10. A | 15. A

What’s In

1. y = -(x − 2)² + 5
   Vertex: (2, 5)
   Axis of symmetry: x = 2
   Graph:

[chart: parabola graph on a coordinate grid; no number or title printed]

2. y = −2(x + 3/2)² + 15/2
   Vertex: (−3/2, 15/2)
   Axis of symmetry: x = −3/2
   Graph:

[chart: parabola graph on a coordinate grid; no number or title printed]

3. y = 4(x − 3/2)² − 2
   Vertex: (3/2, −2)
   Axis of symmetry: x = 3/2
   Graph:

[chart: parabola graph on a coordinate grid; no number or title printed]

4. y = −(x − 3/2)² + 17/4
   Vertex: (3/2, 17/4)
   Axis of symmetry: x = 3/2
   Graph:

[chart: parabola graph on a coordinate grid; no number or title printed]