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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. Power Functionspage 236 What happens if we multiply power functions by constants (a>0, a<0)?

  3. Polynomial Functionspage 236 Every polynomial function is the sum of one or more power functions. Every polynomial function can be expressed in the form: f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 where n is a nonnegative integer an , an-1 , . . . a2 , a1 ,a0 are constants (coefficients) with a0≠ 0 n (the highest power that appears) is called the degree leading term is the term with the highest degree leading coefficient is the coefficient of the leading term examples/page 236

  4. Polynomial Functionspage 236 Every polynomial function is the sum of one or more power functions. Every polynomial function can be expressed in the form: f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Constant Functionsf(x) = kdegree 0 polynomials Linear Functionsf(x) = mx + bfirst degree polynomials

  5. Polynomial Functionspage 236 Every polynomial function can be expressed in the form: f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Section 5.1 Quadratic Functions f(x) = ax2 + bx + c second degree polynomials one turning point is called the VERTEX. BEHAVIOR is up/vertex/down OR down/vertex/up

  6. Suppose an egg is thrown directly upward from a second story window. The initial velocity is 19 feet per second and the height of the window is 20 feet. The model for the egg’s height t seconds after release is given by: f(t) = -16t2 + 19t + 20 initial height acceleration due to gravity initial velocity

  7. Suppose an egg is thrown directly upward from a second story window. The initial velocity is 19 feet per second and the height of the window is 20 feet. The model for the egg’s height t seconds after release is given by: f(t) = -16t2 + 19t + 20 When does the egg hit the ground? How high does the egg go? up/vertex/down What is the velocity of the egg when it hits the ground? parabola opening downward What is the velocity of the egg at its maximum height?

  8. Suppose an egg is thrown directly upward from a second story window. The initial velocity is 19 feet per second and the height of the window is 20 feet. The model for the egg’s height t seconds after release is given by: f(t) = -16t2 + 19t + 20 When does the egg hit the ground? f(t) = 0factor or quadratic formula How high does the egg go? y coordinate of vertex What is the velocity of the egg when it hits the ground? rate of change What is the velocity of the egg at its maximum height? rate of change

  9. Quadratic Functions f(x) = ax2 + bx + c Leading term determines global behavior (as power function). To find y intercept, determine f(0) = c. To find x intercepts of the graph of y=f(x) [or to find zeros of f or to find roots of f(x) = 0], solve f(x) = 0 by factoring or quadratic formula. The axis of symmetry (mid-line) is given by x = -b/2a. The coordinates of the vertex are (-b/2a, f(-b/2a)). The number –b/2a tells where to find the greatest or least valueandf(-b/2a) is that greatest or least value.

  10. Quadratic Functions f(x) = ax2 + bx + c CYU 5.1/page228 f(t) = -16t2 + 19t + 50 CYU 5.2/page233 f(t) = (5/3)t2 – 10t + 45

  11. 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 #5, #7 on page 255

  12. 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

  13. Higher Degree Polynomials 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 … xk zeroes of f. [possibly] more turning points. Identify turning points approximately by graph. (no nice formula)

  14. 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

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

  16. HW Page 255 #1-32 TURN IN: #6,8,16, 24(Maple graph), 26(Maple graph), 32

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