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Nonlinear Functions and their Graphs

Nonlinear Functions and their Graphs. Lesson 4.1. 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”. Polynomial Properties.

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Nonlinear Functions and their Graphs

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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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