holt algebra 1 9-4 transforming quadratic functions warm up for each quadratic function, find the...

Holt Algebra 1

9-4 Transforming Quadratic Functions

Warm 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



Holt Algebra 1

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


Order the functions from narrowest graph to widest.

f(x) = 3x2, g(x) = 0.5x2

Step 1 Find |a| for each function.

|3| = 3 |0.05| = 0.05

Step 2 Order the functions.

f(x) = 3x2

g(x) = 0.5x2

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

Holt Algebra 1


Example 1B: Comparing Widths of Parabolas

Order the functions from narrowest graph to widest.

f(x) = x2, g(x) = x2, h(x) = –2x2

Step 1 Find |a| for each function.

|1| = 1 |–2| = 2

Step 2 Order the functions.

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

f(x) = x2

h(x) = –2x2

g(x) = x2

Holt Algebra 1


Holt Algebra 1


Example 2A: Comparing Graphs of Quadratic Functions

Compare the graph of the function with the graph of f(x) = x2

.

Method 1 Compare the graphs.

• The graph of g(x) = x2 + 3

is wider than the graph of f(x) = x2.

g(x) = x2 + 3

• The graph of g(x) = x2 + 3

opens downward and the graph off(x) = x2 opens upward.

Holt Algebra 1


Example 2B: Comparing Graphs of Quadratic Functions

Compare the graph of the function with the graph of f(x) = x2

g(x) = 3x2

Method 2 Use the functions.• Since |3| > |1|, the graph of g(x) = 3x2 is narrower than the graph of f(x) = x2.

• Since for both functions, the axis of

symmetry is the same.

• The vertex of f(x) = x2 is (0, 0). The vertex of g(x) = 3x2 is also (0, 0).

• Both graphs open upward.

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.

Holt Algebra 1


Example 3: Application

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.

Step 1 Write the height functions. The y-intercept c represents the original height.

h1(t) = –16t2 + 400 Dropped from 400 feet.

h2(t) = –16t2 + 324 Dropped from 324 feet.

Holt Algebra 1


Step 2 Set the equation equal to zero to find the time and solve for t.

0= –16t2 + 400 Dropped from 400 feet.

0= –16t2 + 324 Dropped from 324 feet.

The softball dropped from 400 feet reaches the ground in 5 seconds. The ball dropped from 324 feet reaches the ground in 4.5 seconds

Holt Algebra 1


Holt Algebra 1


Lesson Quiz: Part I

1. Order the function f(x) = 4x2, g(x) = –5x2, and h(x) = 0.8x2 from narrowest graph to widest.

2. Compare the graph of g(x) =0.5x2 –2 with the graph of f(x) = x2.

g(x) = –5x2, f(x) = 4x2, h(x) = 0.8x2

• The graph of g(x) is wider.• Both graphs open upward.• Both have the axis of symmetry x = 0.• The vertex of g(x) is (0, –2); the vertex

of f(x) is (0, 0).

Holt Algebra 1


Lesson Quiz: Part II Two identical soccer balls are dropped. The first is dropped from a height of 100 feet and the second is dropped from a height of 196 feet.

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

The graph of h1(t) = –16t2 + 100 is a vertical translation of the graph of h2(t) = –16t2 + 196 the y-intercept of h1 is 96 units lower than that of h2.

4. Use the graphs to tell when each soccer ball reaches the ground.

2.5 s from 100 ft; 3.5 from 196 ft

Holt Algebra 1


Warm-Up

1. Order the function f(x) = 6x2, g(x) = –3x2, and h(x) = 0.2x2 from narrowest graph to widest.

2. Compare the graph of g(x) =x2 +2 with the graph of f(x) = x2.

holt algebra 1 9-4 transforming quadratic functions warm up for each quadratic function, find the...

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