1. Nonlinear Functions and their Graphs Lesson 4.1

2. Polynomials General formula a 0, a 1, …,a n are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x a n x n 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 a n x n takes over Compare f(x) = x 3 with g(x) = x 3 + x 2 Look at tables Use standard zoom, then zoom out

4. Increasing, Decreasing Functions An increasing functionA decreasing 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(x 3 – 9x 2 ) 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. Try It Out Find local extrema … absolute extrema

11. Assignment Lesson 4.1A Page 256 Exercises 1 – 45 odd

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

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

14. 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

15. Applications Consider the U.S. birthrate from 1900 to 2005 (births per 1000 people) Can be modeled by where x = number of years since 1900 Evaluate f(95) What does it mean? With domain 1900 ≤ x ≤ 2005 Identify the absolute minimum and maximum

16. Applications U.S. natural gas consumption from 1965 to 1980 can be modeled by x = 6 is 1966 and x = 20 is 1980 Consumption measured in trillion cubic feet Evaluate f(10) …. What does it mean? Graph for 6 ≤ x ≤ 20 and 0.4 ≤ y ≤ 0.8 Determine local extrema, interpret results

17. Assignment Lesson 4.1B Page 258 Exercises 91 – 97 odd