The parent quadratic function is f (x) = x2. Its graph is the parabola shown. The axis of symmetry is x = 0. The vertex is (0, 0).

x

f(x)

�2 2O

4

2Vertex (0, 0) Axis of Symmetry

x � 0

Key Concept The Parent Quadratic Function

TEKS (4)(B) Write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.

TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Additional TEKS (1)(E), (1)(F), (1)(G)

TEKS FOCUS

•Maximum value – the greatest y-value of a function

•Minimum value – the least y-value of a function

•Parabola – the set of points in a plane that are the same distance from a fixed point, the focus, as they are from a line, the directrix

•Quadratic function – a function that you can write in the form f(x) = ax2 + bx + c

•Vertex form – The vertex form of a quadratic function is f(x) = a(x - h)2 + k, where a ≠ 0 and (h, k) are the coordinates of the vertex of the function.

•Vertex of a parabola – the point where the function for the parabola reaches a maximum or a minimum value

•Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated

•Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.

VOCABULARY

The graph of any quadratic function is a transformation of the graph of the parent quadratic function, y = x2.

ESSENTIAL UNDERSTANDING

5-1 Attributes and Transformations of Quadratic Functions

152 Lesson 5-1 Attributes and Transformations of Quadratic Functions

2�2

�2

2y

x

2�2

4

2

O

y

x 2�2

4

2

O

y

x

y � x2

y � 2x2

y � �x2 y � x212

Reflection,a and – a

Stretch,a � 1

Compression,0 � a � 1

If a 7 0, the parabola opens upward. The y-coordinate of the vertex is the minimum value of the function.

If a 6 0, the parabola opens downward. The y-coordinate of the vertex is the maximum value of the function.

Maximum Value

y � �x2

Minimum Value

y � x2

Vertex

Key Concept Reflection, Stretch, and Compression

x OO

yy

x O

y

x

y � x2 y � x2

∣h∣

∣k∣∣k∣y � x2y � (x � h)2

y � (x � h)2 � ky � x2 � k

Horizontal Vertical Horizontal and Vertical

Move∣k∣ units.

Move ∣h∣ units.

Move∣h∣ units. Move

∣k∣ units.

∣h∣

vertex becomes (h, 0) vertex becomes (0, k) vertex becomes (h, k)

Key Concept Translation of the Parabola

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Problem 2

Problem 1

Graphing a Function of the Form f(x) = ax2

What is the graph of f (x) = 12x2?

Step 1 Plot the vertex (0, 0). Draw the axis of symmetry, x = 0.

Step 2 Find and plot two points on one side of the axis of symmetry.

Step 3 Plot the corresponding points on the other side of the axis of symmetry.

Step 4 Sketch the curve.

TEKS Process Standard (1)(D)

x

0 (0, 0)12 (0)2 � 0

f(x) � x212

12 (2)2 � 22 (2, 2)

12 (4)2 � 84 (4, 8)

x, f(x)

x

f(x)

�2�4 2 4O

6

4

(4, 8)

(2, 2)(0, 0)

(�2, 2)

(�4, 8)

2 2

4 4

Vertex (0, 0)

Axis of Symmetry x � 0

Graphing Translations of f(x) = x2

Graph each function. How is each graph a translation of f (x) = x2?

A g(x) = x2 − 5

B h(x) = (x − 4)2

Translate the graph of f to the right 4 units to get the graph of h(x) = (x - 4)2.

�2

4

2

�4

y

xO

Vertex(0, 0)

Vertex(0, �5)

f(x) � x2

g(x) � x2 � 5

Axis ofSymmetryx � 0

Translate the graph of f down 5 units to get the graph of g(x) = x2 - 5.

2 4 6�2

�2

�4

2

4y

xO

f(x) � x2 h(x) � (x � 4)2

Axis ofSymmetryx � 4

Vertex(0, 0)

Vertex(4, 0)

How do you choose points to plot?Choose the vertex and two points on one side of the axis of symmetry that give integer values of f(x).

How does g(x) differ from f (x)?For each value of x, the value of g(x) is 5 less than the value of f(x).

154 Lesson 5-1 Attributes and Transformations of Quadratic Functions

Problem 4

Problem 3

Interpreting Vertex Form

For y = 3(x − 4)2 − 2, what are the vertex, the axis of symmetry, the maximum or minimum value, the domain and the range?

Step 1 Compare: y = 3(x - 4)2 - 2 y = a(x - h)2 + k

Step 2 The vertex is (h, k) = (4, -2).

Step 3 The axis of symmetry is x = h, or x = 4.

Step 4 Since a 7 0, the parabola opens upward. k = -2 is the minimum value.

Step 5 Domain: All real numbers. There is no restriction on the value of x. Range: All real numbers Ú -2, since the minimum value of the function is -2.

TEKS Process Standard (1)(G)

Using Vertex Form

A What is the graph of f (x) = −2(x − 1)2 + 3?

B Multiple Choice What steps transform the graph of y = x2 to y = −2(x + 1)2 + 3?

Reflect across the x-axis, stretch by the factor 2, translate 1 unit to the right and 3 units up.

Stretch by the factor 2, translate 1 unit to the right and 3 units up.

Reflect across the x-axis, translate 1 unit to the left and 3 units up.

Stretch by the factor 2, reflect across the x-axis, translate 1 unit to the left and 3 units up.

The correct choice is D.

2

�2 3x

Step 1 Identify the constantsa � �2, h � 1, and k � 3.Because a � 0, the parabolaopens downward.

Step 2 Plot the vertex(h, k) � (1, 3) and draw theaxis of symmetry x � 1.

f(x) � �2(x � 1)2 � 3

Step 4 Sketch the curve.

Step 3 Plot two points. f(2) � �2(2 � 1)2 � 3 � 1.Plot (2, 1) and thesymmetric point (0, 1).

f(x)

How do you use vertex form? Compare y = 3(x - 4)2 - 2 to vertex form y = a(x - h)2 + k to find values for a, h, and k.

What do the values of a, h, and k tell you about the graph?The graph is a stretched reflection of y = x2, shifted 1 unit right and 3 units up.

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1

1234567

2 3 4 5 6 7 8 9 10 11 12 13

Writing a Quadratic Function in Vertex Form

A Nature The picture shows the jump of a dolphin. The vertex of the dolphin’s jump is at the point (3, 7). What quadratic function models the path of the dolphin’s jump?

B A parabola passes through the point (−1, −1), opens upward, and has an axis of symmetry at x = −3 and vertical stretch factor 1. What is the equation of the quadratic function?

Since the parabola opens upward, you know that the value of a is positive, so using the vertical stretch factor, a = 1. Because the axis of symmetry is x = -3, you know that h = -3. Using the vertex form of the quadratic function, you can obtain the equation f (x) = (x + 3)2 + k. Since the graph passes through the point (-1, -1), you can use this to find k.

-1 = f (-1) = (-1 + 3)2 + k

-1 = 22 + k

-1 = 4 + k

-5 = k

The equation of the quadratic function is f (x) = (x + 3)2 - 5.

Problem 5

What is the vertex?

Choose another point, (9, 4), from the path. Substitute in the vertex form.

Solve for a.

Substitute in the vertex form.

The vertex is (3, 7).h = 3, k = 7

f(x) = a(x − h)2 + k 4 = a(9 − 3)2 + 7 4 = 36a + 7 −3 = 36a a = − 1

12

f(x) = − 112(x − 3)2 + 7

models the path of the dolphin’s jump.

PRACTICE and APPLICATION EXERCISES

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M E W O RK

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156 Lesson 5-1 Attributes and Transformations of Quadratic Functions

PRACTICE and APPLICATION EXERCISESON

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HO

M E W O RK

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Graph each function. Describe how it was translated from f (x) = x2.

1. f (x) = x2 + 3 2. f (x) = (x - 2)2 3. f (x) = x2 - 6

4. f (x) = (x + 3)2 5. f (x) = x2 - 9 6. f (x) = (x + 5)2

7. f (x) = x2 + 1.5 8. f (x) = (x - 2.5)2

Identify the vertex, the axis of symmetry, the maximum or minimum value, and the domain and the range of each function.

9. y = -1.5(x + 20)2 10. f (x) = 0.1(x - 3.2)2 11. f (x) = 24(x + 5.5)2

12. y = 0.0035(x + 1)2 - 1 13. f (x) = -(x - 4)2 - 25 14. y = (x - 125)2 + 125

Write a quadratic function to model each graph.

15. 16.

17. Use a Problem-Solving Model (1)(B) A gardener is putting a wire fence along the edge of his garden to keep animals from eating his plants. If he has 20 meters of fence, what is the largest rectangular area he can enclose?

18. You can find the rate of change for an interval between two points of a function by finding the slope between the points. Use the graph to find the y-value for each x-value. Then find the rate of change for each interval.

a. (0, ) and (1, )

b. (1, ) and (2, )

c. (2, ) and (3, )

d. Analyze Mathematical Relationships (1)(F) What do you notice about the rate of change as the interval gets farther away from the vertex?

e. Would your answer to part (d) change if the intervals were on the left side of the graph? Explain.

19. Write a quadratic function to represent the areas of all rectangles with a perimeter of 36 ft. Graph the function and describe the rectangle that has the largest area.

y

2

4

2�4

�2

xO

y2

�4�8

�6

xO

y

2

4

6

8

O 2 4�2�4x

Scan page for a Virtual Nerd™ tutorial video.

157PearsonTEXAS.com

Write the equation of each parabola in vertex form.

20. vertex (0, 5), point (1, -2) 21. vertex (14, -3

2 ), point (1, 3)

22. Create Representations to Communicate Mathematical Ideas (1)(E) Write an equation of a parabola symmetric about x = -10.

23. a. Select Tools to Solve Problems (1)(C) Determine the axis of symmetry for each parabola defined by the spreadsheet values at the right.

b. How could you use the spreadsheet columns to verify that the axes of symmetry are correct?

c. What functions in vertex form model the data?

Check that the axes of symmetry are correct.

Write the equation of each parabola in vertex form.

24. vertex (-7, 6), vertical stretch factor 4, opening upward

25. vertex (23, - 1

3 ), vertical compression factor 13, opening downward

26. axis of symmetry x = 5, vertical stretch factor 8, maximum -2

27. axis of symmetry x = 3.5, vertical compression factor 0.25, minimum -7.5

1

2

3

4

5

6

A

X1

1

2

3

4

5

Y1

�35

�15

�3

1

�3

B

1

2

3

4

5

6

A

X2

1

2

3

4

5

Y2

10

2

2

10

26

B

TEXAS Test Practice

28. One parabola at the right has the equation y = (x - 4)2 + 2. Which equation represents the second parabola?

A. y = -(x - 4)2 + 2 C. y = (x + 4)2 - 2

B. y = (-x - 4)2 + 2 D. y = -(x + 4)2 - 2

29. Which system has the unique solution (1, 4)?

F. e y = x - 3

x + y = 5 H. e

x + y = 5

y = -x + 3

G. e y = -x + 5

x - y = -3 J. e

2x + y = 3

2x - 2y = -6

30. What is the formula for the surface area of a right circular cylinder, S = 2prh + 2pr2, solved for h?

A. h = S4pr B. h = S

2pr2 C. h = S2pr - r D. h = r - S

2pr

2 4 6

�2

4

6

2

�2

y

xO

158 Lesson 5-1 Attributes and Transformations of Quadratic Functions