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

Holt Algebra 1

9-4 Transforming Quadratic FunctionsWarm UpFor each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward.1. y = x2 + 3

2. y = 2x2 3. y = –0.5x2 – 4

x = 0; (0, 3); opens upward

x = 0; (0, 0); opens upward

x = 0; (0, –4); opens downward

Holt Algebra 1

9-4 Transforming Quadratic Functions

Graph and transform quadratic functions.

Objective

Holt Algebra 1


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

Remember!

Holt Algebra 1


The quadratic parent function is f(x) = x2. The graph of all other quadratic functions are transformations of the graph of f(x) = x2.For the parent function f(x) = x2:

• 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


Holt Algebra 1


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 FunctionsOrder the functions from narrowest graph to widest.

f(x) = 3x2, g(x) = 0.5x2 Find |A| for each function.

|3| = 3 |0.05| = 0.05

f(x) = 3x2

g(x) = 0.5x2

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

1.2.

Holt Algebra 1


f(x) = x2, g(x) = x2, h(x) = –2x2

|1| = 1 |–2| = 2

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

f(x) = x2

h(x) = –2x2

g(x) = x2

1.2.3.

Holt Algebra 1


f(x) = –x2, g(x) = x2

|–1| = 1The function with the

narrowest graph has the greatest |A|.

f(x) = –x2

g(x) = x2

1.2.

Holt Algebra 1


Holt Algebra 1


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) = ax2 up or down the y-axis.

Holt Algebra 1


Holt Algebra 1


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

Helpful Hint

Holt Algebra 1

9-4 Transforming Quadratic FunctionsCompare the graph of the function with the graph of f(x) = x2

.

• The graph of g(x) = x2 + 3is wider than the graph of f(x) = x2.

g(x) = x2 + 3

• The graph of g(x) = x2 + 3opens downward. 2f x x

21 34

g x x

Holt Algebra 1

9-4 Transforming Quadratic FunctionsCompare the graph of the function with the graph of f(x) = x2

g(x) = 3x2

2f x x

23g x x

Holt Algebra 1

9-4 Transforming Quadratic FunctionsCompare the graph of each the graph of f(x) = x2.

g(x) = –x2 – 4

2f x x

2 4g x x

Holt Algebra 1

9-4 Transforming Quadratic FunctionsCompare the graph of the function with the graph of f(x) = x2.

g(x) = 3x2 + 9

2f x x

23 9g x x

3

2f x x

Holt Algebra 1

9-4 Transforming Quadratic FunctionsCompare the graph of the function with the graph of f(x) = x2.

g(x) = x2 + 2

2f x x

2 4g x x

x 21 22

g x x

2 4

Wider

Holt Algebra 1


The quadratic function h(t) = –16t2 + 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 FunctionsTwo 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.h1(t) = –16t2 + 400 Dropped from 400 feet. h2(t) = –16t2 + 324 Dropped from 324 feet.

50

2

2

16 400 0

16 400

t

t

16 16

2 255

tt

2

2

16 324 0

16 324

t

t

16 16

2 81/ 49 / 2

tt

h t

t

Holt Algebra 1

9-4 Transforming Quadratic FunctionsThe graph of h2 is a vertical translation of the graph of h1. Since the softball in h1 is dropped from 76 feet higher than the one in h2, the y-intercept of h1 is 76 units higher.

50

h t

t

21 16 400h t t 2

2 16 324h t t

b. Use the graphs to tell when each softball reaches the ground. 4.5 seconds

5 seconds

Holt Algebra 1


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

Caution!

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

Documents

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