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Understanding Polynomials and Nonlinear Functions

Learn about polynomial properties, end behavior, increasing and decreasing functions, extrema, and graphing techniques for nonlinear functions. Discover how to identify local and absolute extrema, as well as even and odd functions.

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Understanding Polynomials and Nonlinear Functions

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  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 • Comparef(x) = x3 with g(x) = x3 + x2 • Look at tables • Use standard zoom, then zoom out

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

  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= 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

  7. • • 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

  8. 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 •

  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

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