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# Quadratic Functions: Parabolas

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 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
(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 = 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?
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.