Tid Bits From Chapter Three

Functions consist of:
1. Domain (x)
2. Each input determines only one output
3. Range (y)

The Vertical Line Test => A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once.

To find the Maximum and Minimum values of a graph:
Step 1: On your calculator press: 2nd and then Trace
Step 2: Finding Minimum (option number 3) or Maximum (option number 4) press either one
Step 3: Right bound, Left bound, Approximate guess as to where the maximum or minimum is, Enter--and you have found the Maximum or Minimum!

Forms of Quadratic Functions:
Transformation form: F(x) = a (x-h)² + K
Polynomial form: F(x) = ax² + bx + c
x-Intercept form: F(x) = a (x-s) (x-t)
All the letters (a, h, k, b, c, s, and t) are real numbers-a does not equal zero. When "a" is positive the graph will open upwards. When "a" is negative the graph will open downwards.

Y =  a (x-h)² + K----> a is the width of the parabola
                                                    -(x-h)² is where the parabola is moving (left or right)
                                                    -k is where the parabola is moving vertically


-When h is negative the parabola moves to the right
-When h is positive the parabola moves to the left

-The vertex is the lowest or highest point in the parabola

The Composite Function of f and g: (g ◦ f) (x) = g (f(x))
Example:
If F(x) = 4x² + 1 and g(x) = 1/x+2 find (g ◦ f)(2).
Step 1: Find f(2) first
f(2) = 4(2)² + 1
f(2) = 17
Step 2: Use the answer to f(2), which is 17, as an input in g
g(17) = 1/17+2
g(17) = 1/19
So (g ◦ f)(2) is equal to 1/19


The Difference Quotient: f(x+h) - f(x)/h