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LINEAR AND QUADRATIC FUNCTIONS

4.2a LINEAR AND QUADRATIC FUNCTIONS

A linear function ,f of x, can be written in the form f(x)=ax+b, where a and b are real numbers. The graph of a linear function is a nonvertical linea with slope a and y -intercept (0,b)

Problem Type #1: Consider the fol lowing linear function. Reduce all fractions to lowest terms.

1) Find the slope and y-intercept (entered as an ordered pair ) of the line which is represented by this function.

2) Graph this linear function by finding two points on the line. (Note: You can use the y- intercept as one of the points .)

EX1:

EX2:

EX3:

A quadratic function ,f of x, can be written in the form where a, b, and c are real numbers and a ≠0. The graph of a quadratic function is a parabola.

The vertex form of a quadratic function is .
(Vertex form can be arrived at by completing the square or by using the Vertex Formula .)

The graph of f is a parabola with vertex (h,k) and axis of symmetryhx x=h

The parabola opens up a>0 if with the function having a minimum value at k.

The parabola a >0 a>0 if with the function having a maximum value at k .

The Vertex Formula
The vertex of the graph of

Problem Type #2 :Consider the following quadratic function
Reduce all fractions to the lowest terms . :

1) Find the vertex of this function.
2) Enter the x-intercept(s), if any, of this function as ordered pair(s).
3) Graph this quadratic function by identifying two points on the parabola other that the vertex and the x-intercept(s).

EX4:

EX5:

EX6:

4.2b MAX/MIN APPLICATIONS OF QUADRATIC FUNCTIONS

Problem Type #1:

EX1:
A small cruising ship that can hold up to 56 people provides three-day excursions to groups of 36 or more. If the group contains 36 people, each person pays $62. The cost per person is reduced by $1 for each person in excess of 36. Find the size of the group that maximizes income for the owners of the ship.

Problem Type #2:

EX2:
A rancher has 400 feet of fencing to put around a rectangular field and then sub divide the field into 3 identical smaller rectangular plots by placing two fences parallel to one of the field’s shorter sides. Find the dimensions that maximize the enclosed area.

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