1. Pre Calculus
Day 5

2. Plan for the day Review of homework Analyzing any graph of a function
1.3 page # 1-9 odd, (odd), 23, 30, 33, 35, (odd), (odd), 77, 78, 80
Analyzing any graph of a function
Domain and Range
Symmetry
Intercepts and zeros
Relative maximums and minimums
Increasing, decreasing and constant intervals
Homework
1.4 page # 1-4, odd, odd, odd

3. Analyzing any Function
Graphs of functions help you visualize relationships between two variables. Finding key information to help describe the graph is analyzing the function. What mathematical terms might help describe a graph?

4. Analyzing Graphs of Functions
Remember that a graph of a function is the collection of ordered pairs (x, f(x)) placed on a coordinate plane. Sometimes the number of ordered pairs are infinite, sometimes they are not. The set of all values for “x” is the domain and it represents the directed distance from the y-axis. The set of all values for “y” is the range and it represents the directed distance from the x-axis. Determining the domain and range is one element in analyzing graphs.

5. Analyzing Graphs of Functions
Graphs may have a shape that can be described.
There are two types of symmetry associated with graphs. Functions containing symmetry have special names:
Even – An even function has line (or reflection) symmetry around the y-axis. An even function is defined such that f(x) = f(-x)
Odd – An odd function has point (or rotation) symmetry about the origin. An odd function is defined such that f(-x) = -f(x)
Describing the shape using symmetry: even, odd or neither is another element in analyzing graphs.

6. Examples of Even and Odd Functions; Symmetry
Even: Odd:

7. Analyzing Graphs of Functions
Graphs may have key points of interest. The y-intercept, x-intercept, any high or low points may be of interest in analyzing the graph. The y-intercept can be found by setting x = 0 and simplifying The x-intercepts, also known as the zeros of the function, can be found by setting f(x) = 0, and solving for x. This may require one of many techniques you have learned (taking square roots, factoring, quadratic formula to name a few) High and low points in terms of maximums, minimums, relative maximums and relative minimums Identifying key points is another element in analyzing graphs.

8. Relative Maximum and Relative Minimum
A function value f(a) is called a relative minimum of f if there exists an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a)  f(x)
A function value f(a) is called a relative maximum of f if there exists an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a)  f(x)

9. Analyzing Graphs of Functions
As you review a graph from left to right on the number line, the changes in the graph should be noted. The intervals of the independent variable “x” can be used to describe the graph or output as increasing, decreasing or constant. Graphs can change direction at its maximums, minimums, relative maximums and relative minimums Identifying increasing, decreasing and constant intervals is element in analyzing graphs.

10. Increasing and Decreasing Functions
A function f is increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2) A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2) A function f is constant on an interval if, for any x1 and x2 in the interval implies f(x1) = f(x2)

11. Analyze the function below:
y = f (x)
(0, 5)
(-1, 0)
(6, 1)
(1, 0)
(5, 0)
(-2, -5)
(4, -5)
(3, -7)

12. Example: Analyzing a function
Type of function
Domain and Range
Relative Minimum and maximum
Intercepts and Zeros
Increasing and Decreasing Intervals
Symmetry
What is the value of f(4)
The value of x at f(x)=-7

13. Homework
Homework #3
1.4 page # 1-4, odd, odd, odd