## Tuesday, September 25, 2007

### Precalculus Chapter 1.1b + 1.2a

Chapter 1.1b Functions + 1.2a Graphs of Functions

I. Application: A right triangle is formed in the first quadrant by the x-axis and y-axis and a line through the point (2, 1). Write the area of the triangle as a function of “x”, and determine the domain of the function.

1. Graph the triangle:
- one side of the triangle is the x-axis and another side is the y-axis. Since we do not know exactly where the points of the third line cross the 2 axes, we use the points (0,y) and (x,0).

A = ½ (base)(height) = ½ xy

Since (0, y) , (2, 1) and (x, 0) all lie on the same line, the slopes between any pairs of points are equal.

M = (1 – y)/(2 – 0) = (1 – 0)/(2 – x)

1 – y = 2/(2 – x)

y = -2/ ( 2 – x) + 1

y = (-2 + (2 – x))/ (2 – x)

y = -x / (2 – x)

y = x / (x – 2)

Now using substitution:

A = ½ x (x / (x – 2)) = x2/(2x – 4)

The domain is (2, ∞ ) since the area has to be greater than zero.

II. Evaluating a Difference Quotient:

If f (x) = 2x, then using the difference quotient below

Using f(x) = 2x, plug into the equation above:

2h/h = 2

Example 2: Given 5x – x2,

-5 – h , where h cannot equal zero

Example 3:

Increasing, Decreasing, and Constant Functions

A function “f” is increasing on an interval if, for any x1 and x2 in the interval, when x1 is less than x2 implies that f (x1 ) is less than f(x2 )
A function “f” is decreasing on an interval if, for any x1 and x2 in the interval, when x1 is less than x2 implies that f (x1 ) is greater than f(x2 )
A function “f” is constant on an interval if, for any x1 and x2 in the interval, f (x1 ) = f(x2 )

Check out this website:
http://www.mathematicshelpcentral.com/lecture_notes/precalculus_algebra_folder/increasing_and_decreasing_functions.htm

Relative Minimum and Relative Maximum:
1. A function value f (a) is called a relative minimum of “f” if there exists an interval that contains a such that:

So you can see that the function is increasing from (- ∞ , -3) and then again from (2, ∞). The function is decreasing from (-3, 2)

Now lets see an example of relative minimum and relative maximum points:

So you can see that the function's relative minimum point from ( - ∞, 0 ) is
"a" while its relative minimum point from ( 0, ∞) is "c" . The relative maximum point is "b".