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
is the optimum value (either maximum or minimum) of the function.
* Zeros of quadratic functions = Solutions of the quadratic equation ax2 + bx + c = 0
* y-intercept = y value when x = 0
Ex.1 (#6) f(x) = x2 + 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
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.
Ex.3 How is the graph of y = x2 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?
(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.