Nonlinear Functions and their Graphs

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. • • Extrema of Nonlinear Functions • Given the function for the Y= screeny1(x) = 0.1(x3 – 9x2) • Use window -10 < x < 10 and -20 < y < 20 • Note the "top of the hill" and the"bottom of thevalley" • These are localextrema

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

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

6. 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

7. Try It Out • Find local extrema … absolute extrema

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

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

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

11. 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

12. 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

13. 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

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