**Definitions:** Quadratic functions:

where a, b, c are real numbers.

**The Graph of a Quadratic Function**

* Shape: parabola, symmetric

When a > 0

When a < 0

* Vertex:

is the optimum value (either maximum or minimum) of the function.

* Zeros of quadratic functions = Solutions of the quadratic equation ax^{2} + bx
+ c = 0

* y-intercept = y value when x = 0

Ex.1 (#6) f(x) = x^{2} + 2x - 3

(a) The vertex

(b) Is the vertex a maximum or minimum point?

(c) x value for the optimal value

(d) The optimal value

(e) Zeros

(f) y-intercept

(g) Graph

Ex.2 (# 36) If 100 feet of fence is used to enclose a rectangular yard,
determine the length and width of

the rectangle that give maximum area.

**Shifting Graphs**

Compare vertices.

Ex.3 How is the graph of y = x^{2} shifted to get the graph of

Ex.4 How is the graph of shifted to get the graph of

**Average Rate of Change**

The rate of change of a quadratic function is not constant.

-> use the **average** rate of change between two points.

Definition: The average rate of change of f(x) between two points x = a and x = b (a < b) is

Average rate of change

= Slope of the line between (a, f(a)) and (b, f(b))

Ex.5 (#42) The figure (on p.160) shows the graph of a total revenue function,
with x equal to the

number of units sold.

(a) Is the average rate of change of revenue negative from x = a to x = b or
from x = b to x = c?

Explain.

(b) Would the number of units d need to satisfy d < b or d > b for the
average rate of change of

revenue from x = a to x = d to be greater than that from x = a to x = b?
Explain.