1. Nonlinear Functions and their Graphs
Lesson 4.1

2. Polynomials General formula a0, a1, … ,an are constant coefficients
n is the degree of the polynomial
Standard form is for descending powers of x
anxn is said to be the “leading term”

3. Polynomial Properties
Consider what happens when x gets very large negative or positive
Called “end behavior”
Also “long-run” behavior
Basically the leading term anxn takes over
Compare f(x) = x3 with g(x) = x3 + x2
Look at tables
Use standard zoom, then zoom out

4. Increasing, Decreasing Functions
A decreasing function
An increasing function

5. Increasing, Decreasing Functions
Given Q = f ( t )
A function, f is an increasing function if the values of f increase as t increases
The average rate of change > 0
A function, f is an decreasing function if the values of f decrease as t increases
The average rate of change < 0

6. Extrema of Nonlinear Functions
Given the function for the Y= screen y1(x) = 0.1(x3 – 9x2)
Use window -10 < x < 10 and -20 < y < 20
Note the "top of the hill" and the "bottom of the valley"
These are local extrema •

7. Extrema of Nonlinear Functions•
Local maximum
f(c) ≥ f(x) when x is near c
Local minimum
f(n) ≤ f(x) when x is near n c n

8. Extrema of Nonlinear Functions
Absolute minimum
f(c) ≤ f(x) for all x in the domain of f
Absolute maximum
f(c) ≥ f(x) for all x in the domain of f
Draw a function with an absolute maximum •

9. Extrema of Nonlinear Functions
The calculator can find maximums and minimums
When viewing the graph, use the F5 key pulldown menu
Choose Maximum or Minimum
Specify the upper and lower bound for x (the "near")
Note results

10. Assignment
Lesson 4.1A
Page 232
Exercises 1 – 45 odd

11. Even and Odd Functions
If  f(x) = f(-x)  the graph is symmetric across the y-axis
It is also an even function

12. Even and Odd Functions
If f(x) = -f(x) the graph is symmetric across the x-axis
But ... is it a function ??

13. Even and Odd Functions A graph can be symmetric about a point
Called point symmetry
If f(-x) = -f(x) it is symmetric about the origin
Also an odd function

14. Assignment
Lesson 4.1B
Page 234
Exercises 45 – 69 odd