1. Warm Up
For 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

2. 9-4
Transforming Quadratic Functions
Holt Algebra 1

3. The quadratic parent function is f(x) = x2
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.

5. 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|.

6. 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.
h(x) = –2x2
The function with the narrowest graph has the greatest |a|.
f(x) = x2
g(x) = x2

8. Example 2A: Comparing Graphs of Quadratic Functions
Compare the graph of the function with the graph of f(x) = x2.
g(x) = x2 + 3
Method 1 Compare the graphs.
The graph of
g(x) = x2 + 3
is wider than the graph of
f(x) = x2.
The graph of
g(x) = x2 + 3
opens downward and the graph of
f(x) = x2 opens upward.

9. 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.

10. 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.

11. 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) = –16t Dropped from 400 feet.
h2(t) = –16t Dropped from 324 feet.

12. Step 2 Set the equation equal to zero to find the time and solve for t.
0= –16t Dropped from 400 feet.
0= –16t 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

14. 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).

15. 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) = –16t is a vertical translation of the graph of h2(t) = –16t 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

16. 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.