College Algebra 9th txtbk

SECTION 3–4 Quadratic Functions 205
We’ll see where the name comes from in a bit. For now, refer to Explore-Discuss 1. Any
function of this form is a transformation of the basic squaring function g(x) x 2 , so we
can use transformations to analyze the graph.
EXAMPLE 1
The Graph of a Quadratic Function
Use transformations of g(x) x 2 to graph the function f (x) 2(x 3) 2 4. Use your
graph to determine the graphical significance of the constants 2, 3, and 4 in this
function.
SOLUTION
Multiplying by 2 vertically stretches the graph by a factor of 2. Subtracting 3 inside the square
moves the graph 3 units to the right. Adding 4 outside the square moves the graph 4 units up.
The graph of f is shown in Figure 3, along with the graph of g(x) x 2 .
y
y x 2 y 2(x 3) 2 4
10
5
(3, 4)
5
5
x
Z Figure 3
The lowest point on the graph of f is (3, 4), so h 3 and k 4 determine the key point
where the graph changes direction. The constant a 2 affects the width of the parabola.
MATCHED PROBLEM 1 Use transformations of g(x) x 2 to graph the function f (x) 1 2(x 2) 2 5. Use your
graph to determine the significance of the constants 1 2, 2, and 5 in this function.
Every parabola has a point where the graph reaches a maximum or minimum and changes
direction. We will call that point the vertex of the parabola. Finding the vertex is key to
many of the things we’ll do with parabolas. Example 1 and Explore-Discuss 1 demonstrate
that
if a quadratic function is in the form f (x) a(x h) 2 k,
then the vertex is
the point (h, k).
Next, notice that the graph of h(x) x 2 is symmetric about the y axis. As a result, the
transformation f (x) 2(x 3) 2 4 is symmetric about the vertical line x 3 (which runs
through the vertex). We will call this vertical line of symmetry the axis, or axis of symmetry
of a parabola. If the page containing the graph of f is folded along the line x 3,
the two halves of the graph will match exactly.
Finally, Explore-Discuss 1 illustrates the significance of the constant a in
f (x) a(x h) 2 k. If a is positive, the graph has a minimum and opens upward. But if
a is negative, the graph will be a vertical reflection of h(x) x 2 and will have a maximum
and open downward. The size of a determines the width of the parabola: if a 7 1, the
graph is narrower than h(x) x 2 , and if a 6 1, it is wider.
These properties of a quadratic function in vertex form are summarized next.