Quadratic Functions ; maxima and Minima

Graphing Quadratic Functions Using the Standard Form

A quadratic function is a function f of the form
where a, b, and c are real numbers and a ≠0.

For example, if we take a=1,b=c=0, we get the simple quadratic function
The graph of any quadratic function is called a parabola, and can be obtained from the graph of by the transformations discussed in § 2.4.

Standard Form of a Quadratic Function
A quadratic function can be ex pressed in the standard form

by completing the square . The graph of f is a parabola with vertex (h, k); the parabola opens upward if a>0 or downward if a<0.

Example 1
Standard Form of a Quadratic Function
Let
(a) Express f in standard form.
(b) Sketch the graph of f

Maximum and Minimum Values of Quadratic Functions

Maximum or Minimum Value of a Quadratic Function
Let f be a quadratic function with standard form The maximum or minimum value of f occurs at x=h.
If a>0, then the minimum value of f is f (h) =k.
If a<0, then the maximum value of f is f (h) =k.

Example 2
Minimum Value of a Quadratic Function

Consider the quadratic function
(a) Express f in standard form.
(b) Find the minimum value of f.

Example 3
Maximum Value of a Quadratic Function

Consider the quadratic function
(a) express f in standard form.
(b) Find the maximum value of f.

We now derive a formula for the maximum or minimum of the quadratic function in terms of a , b, and c.

Factor a from the x -terms
Complete the square.
Factor and simplify

Thus we get

Maximum of Minimum Value of a Quadratic Function
The maximum or minimum value of a quadratic function f(x) =ax²+bx+c occurs at

 the maximum or minimum value is


Example 4
Finding Maximum and Minimum Values of Quadratic Functions

Find the maximum or minimum value of each quadratic function.
(a) f(x) =x²+4x
(b) f(x) =−2x²+4x−5

Example 5
Advertising

The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments and advertising agency found that if the effectiveness E is measured on a scale of 1 to 10, then

where n is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?


Example 6
Domain and Range of Quadratic Functions

Find the domain and range of each of the fol lowing quadratic functions.
(a) f(x) =−x²+4x−3
(b) f(x) =2x²+6x−7