1. Objective
Graph and transform quadratic functions.

2. 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.

4. 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

5. Example 1B Continued
Order the functions from narrowest graph to widest.
f(x) = x2, g(x) = x2, h(x) = –2x2
Check Use a graphing calculator to compare the graphs.
h(x) = –2x2 has the narrowest graph and 
g(x) = x2 has the widest graph.

6. Check It Out! Example 1b
Order the functions from narrowest graph to widest.
f(x) = –4x2, g(x) = 6x2, h(x) = 0.2x2
Step 1 Find |a| for each function.
|–4| = 4
|6| = 6
|0.2| = 0.2
Step 2 Order the functions.
g(x) = 6x2
The function with the narrowest graph has the greatest |a|.
f(x) = –4x2
h(x) = 0.2x2

7. Check It Out! Example 1b Continued
Order the functions from narrowest graph to widest.
f(x) = –4x2, g(x) = 6x2, h(x) = 0.2x2
Check Use a graphing calculator to compare the graphs.
g(x) = 6x2 has the narrowest graph and 
h(x) = 0.2x2 has the widest graph.

9. 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.

10. Example 2A Continued
Compare the graph of the function with the graph of f(x) = x2
g(x) = x2 + 3
The vertex of
f(x) = x2 is (0, 0).
g(x) = x2 + 3
is translated 3 units up to (0, 3).
The axis of symmetry is the same.

11. Check It Out! Example 2c
Compare the graph of the function with the graph of f(x) = x2.
g(x) = x2 + 2
Method 1 Compare the graphs.
The graph of
g(x) = x2 + 2
is wider than the graph of
f(x) = x2.
The graph of
g(x) = x2 + 2
opens upward and the graph
of f(x) = x2 opens upward.

12. Check It Out! Example 2c Continued
Compare the graph of the function with the graph of f(x) = x2.
g(x) = x2 + 2
The vertex of
f(x) = x2 is (0, 0).
g(x) = x2 + 2
is translated 2 units up to (0, 2).
The axis of symmetry is the same.