9-4 Transforming Quadratic Functions Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward.

1. y = x 2 + 3 x = 0; (0, 3); opens upward 2. y = 2x 2 x = 0; (0, 0); opens upward 3. y = –0.5x 2 – 4 x = 0; (0, –4); opens downward

Holt Algebra 1

9-4 Transforming Quadratic Functions

Learning Target

Students will be able to: Graph and transform quadratic functions.

Holt Algebra 1

9-4 Transforming Quadratic Functions Remember!

You saw in Lesson 5-9 that the graphs of all linear functions are transformations of the linear parent function y = x.

Holt Algebra 1

9-4 Transforming Quadratic Functions

The quadratic parent function is f(x) = x

2

. The graph of all other quadratic functions are transformations of the graph of f(x) = x 2 .

For the parent function f(x) = x 2 : • The axis of symmetry is x = 0, or the y-axis.

• The vertex is (0, 0) • The function has only one zero, 0 .

Holt Algebra 1

9-4 Transforming Quadratic Functions Holt Algebra 1

9-4 Transforming Quadratic Functions

The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

Holt Algebra 1

9-4 Transforming Quadratic Functions Order the functions from narrowest graph to widest.

f(x) = 3x 2 , g(x) = 0.5x 2

Find |A| for each function.

|3| = 3 |0.05| = 0.05

The function with the narrowest graph has the greatest |A|.

1.

f(x) = 3x 2 2.

g(x) = 0.5x 2

Holt Algebra 1

9-4 Transforming Quadratic Functions Order the functions from narrowest graph to widest.

f(x) = x 2 , g(x) = x 2 , h(x) = –2x 2

|1| = 1 |–2| = 2 1.

2.

3.

h(x) = –2x 2 f(x) = x 2 g(x) = x 2

The function with the narrowest graph has the greatest |A|.

Holt Algebra 1

9-4 Transforming Quadratic Functions Order the functions from narrowest graph to widest.

f(x) = –x 2 , g(x) = x 2

The function with the

|–1| = 1

narrowest graph has the greatest |A|.

1.

2.

f(x) = –x 2 g(x) = x 2

Holt Algebra 1

9-4 Transforming Quadratic Functions Holt Algebra 1

9-4 Transforming Quadratic Functions

The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-intercept but also a vertical translation of the graph of f(x) = ax 2 up or down the y-axis.

Holt Algebra 1

9-4 Transforming Quadratic Functions Holt Algebra 1

9-4 Transforming Quadratic Functions Helpful Hint

When comparing graphs, it is helpful to draw them on the same coordinate plane.

Holt Algebra 1

9-4 Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2 .

• The graph of g(x) = x 2 + 3

g(x) = x 2 + 3

is wider than the graph of f(x) = x 2 .

• The graph of g(x) = x 2 + 3 opens downward .



x

2   1 4

x

2  3

Holt Algebra 1

9-4 Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2 g(x) = 3x 2

 3

x

2 

x

2

Holt Algebra 1

9-4 Transforming Quadratic Functions Compare the graph of each the graph of f(x) = x 2 .

g(x) = –x 2 – 4



x

2 4

Holt Algebra 1

9-4 Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x g(x) = 3x 2 2 .

+ 9

   3

x

2  9  3

Holt Algebra 1

9-4 Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2 .

g(x) = x 2 + 2



Wider

4 

x

2

x

 1 2

x

2  2  2 4

Holt Algebra 1

9-4 Transforming Quadratic Functions

The quadratic function h(t) = –16t 2 + c can be used to approximate the height h in feet above the ground of a falling object t seconds after it is dropped from a height of c feet. This model is used only to approximate the height of falling objects because it does not account for air resistance, wind, and other real-world factors.

Holt Algebra 1

9-4 Transforming Quadratic Functions Two identical softballs are dropped. The first is dropped from a height of 400 feet and the second is dropped from a height of 324 feet.

a. Write the two height functions and compare their graphs.

h

1 (t) = –16t 2 + 400

Dropped from 400 feet.

h

2 (t) = –16t 2  16

t

2  400  0 + 324

Dropped from 324 feet.

 16

t

2  324  0     16 16

t

2

t

2   400  16  25

t

 5   16 16

t

2

t

2

t

  324  16  81/ 4  9 / 2 50

t

Holt Algebra 1

9-4 Transforming Quadratic Functions

The graph of h 2 graph of h 1 is a vertical translation of the . Since the softball in h 1 is dropped from 76 feet higher than the one in h intercept of h 1 is 76 units higher.

2 , the y         16

t

2   16

t

2   400 324 50

t

b. Use the graphs to tell when each softball reaches the ground.

4.5 seconds 5 seconds

Holt Algebra 1

9-4 Transforming Quadratic Functions Caution!

Remember that the graphs show here represent the height of the objects over time, not the paths of the objects.

HW pp. 617-619/10-42, 44-49

Holt Algebra 1