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Chapter 5 Polynomial and Rational Functions

Chapter 5 Polynomial and Rational Functions. 5.1 Quadratic Functions and Models 5.2 Polynomial Functions and Models 5.3 Rational Functions and Models. A linear or exponential or logistic model either increases or decreases but not both.

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Chapter 5 Polynomial and Rational Functions

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  1. Chapter 5 Polynomial and Rational Functions 5.1 Quadratic Functions and Models 5.2 Polynomial Functions and Models 5.3 Rational Functions and Models A linear or exponential or logistic model either increases or decreases but not both. Life, on the other hand gives us many instances in which something at first increases then decreases or vice-versa. For situations like these, we might turn to polynomial models.

  2. Quadratic Functions f(x) = ax2 + bx + c CYU 5.2/page233 f(t) = (5/3)t2 – 10t + 45

  3. Quadratic Functions f(x) = ax2 + bx + c FACTORED FORM: f(x) = a(x-x1)(x-x2) for x1 and x2 zeroes of f. VERTEX FORM: f(x) = a(x-h)2 + k for vertex (h,k). All quadratic functions are tranformations of f(x) = x2 CYU 5.3/page 234

  4. Optimization(finding maximum/minimum values in context) (page232) Suppose Jack has 188 feet of fencing to make a rectangular enclosure for his cow. Find the dimensions of the enclosure with maximum area. Build an area function and find maximum value. More Practice #15/257

  5. Higher Degree Polynomials (5.2) f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Graph is always a smooth curve Leading term determines global behavior (as power function). To find y intercept, determine f(0) = c. To find x intercepts, solve f(x) = 0 by factoring or SOLVE command. FACTORED FORM: f(x) = a(x-x1)(x-x2)…(x-xk) for x1, x2 … xkzeroes of f. [possibly] more turning points. Identify turning points approximately by graph. (no nice formula)

  6. Higher Degree Polynomials f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 The speed of a car after t seconds is given by: f(t) = .005t3 – 0.21t2 + 1.31t + 49 (3.46, 51.27) extended view global behavior matches leading term (24.44, 28.35) local maximum and local minimum CYU 5.5/240

  7. The speed of a car (in mph) after t seconds is given by: f(t) = .005t3 – 0.21t2 + 1.31t + 49 According to Maple t-intercept is -11.34 rate of change at t = 15is -1.625 rate of change at t = 26 is 0.52 (3.46, 51.27) (24.44, 28.35) These calculations agree with the graph, since slope of curve is negative at t = 15 and positive at t = 26.

  8. Higher Degree Polynomials f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Find a formula for a polynomial whose graph is shown below.

  9. HW Page 255 #1-32 TURN IN: #16, 24(Maple graph), 26(Maple graph), 32

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