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 Functions page 236 DomainRangeGlobal Behavior Turning Points x0x0 R{1}Level0 x1x1 RRDown/Up0 x2x2 RNonnegative reals Up/Up1 x3x3 RRDown/Up0 x4x4 RNonnegative Reals Up/Up1 x5x5 RRDown/Up0 x6x6 RNonnegative Reals Up/Up1 x7x7 RRDown/Up0 What happens if we multiply power functions by constants (a>0, a<0)?

3. Polynomial Functions page 236 Every polynomial function is the sum of one or more power functions. Every polynomial function can be expressed in the form: f(x) = a n x n + a n-1 x n-1 +... a 2 x 2 + a 1 x + a 0 where n is a nonnegative integer a n, a n-1,... a 2, a 1,a 0 are constants (coefficients) with a 0 ≠ 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 Functions page 236 Every polynomial function is the sum of one or more power functions. Every polynomial function can be expressed in the form: f(x) = a n x n + a n-1 x n-1 +... a 2 x 2 + a 1 x + a 0 Constant Functions f(x) = k degree 0 polynomials Linear Functions f(x) = mx + b first degree polynomials

5. Polynomial Functions page 236 Every polynomial function can be expressed in the form: f(x) = a n x n + a n-1 x n-1 +... a 2 x 2 + a 1 x + a 0 Quadratic Functions f(x) = ax 2 + bx + c second degree polynomials one turning point is called the VERTEX. BEHAVIOR is up/vertex/down OR down/vertex/up Section 5.1

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) = -16t 2 + 19t + 20 acceleration due to gravity initial velocity initial height

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) = -16t 2 + 19t + 20 up/vertex/down parabola opening downward When does the egg hit the ground? How high does the egg go? What is the velocity of the egg when it hits the ground? 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) = -16t 2 + 19t + 20 When does the egg hit the ground? f(t) = 0 factor 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) = ax 2 + bx + c 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 value and f(-b/2a) is that greatest or least value. Leading term determines global behavior (as power function).

10. Quadratic Functions f(x) = ax 2 + bx + c CYU 5.1/page228 f(t) = -16t 2 + 19t + 50 CYU 5.2/page233 f(t) = (5/3)t 2 – 10t + 45

11. Quadratic Functions f(x) = ax 2 + bx + c VERTEX FORM: f(x) = a(x-h) 2 + k for vertex (h,k). All quadratic functions are tranformations of f(x) = x 2 FACTORED FORM: f(x) = a(x-x 1 )(x-x 2 ) for x 1 and x 2 zeroes of f. 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) = a n x n + a n-1 x n-1 +... a 2 x 2 + a 1 x + a 0 To find y intercept, determine f(0) = c. To find x intercepts, solve f(x) = 0 by factoring or SOLVE command. Leading term determines global behavior (as power function). [possibly] more turning points. Identify turning points approximately by graph. (no nice formula) FACTORED FORM: f(x) = a(x-x 1 )(x-x 2 )…(x-x k ) for x 1, x 2 … x k zeroes of f. Graph is always a smooth curve

14. Higher Degree Polynomials f(x) = a n x n + a n-1 x n-1 +... a 2 x 2 + a 1 x + a 0 The speed of a car after t seconds is given by: f(t) =.005t 3 – 0.21t 2 + 1.31t + 49 local maximum and local minimum extended view global behavior matches leading term (3.46, 51.27) (24.44, 28.35)

15. Higher Degree Polynomials f(x) = a n x n + a n-1 x n-1 +... a 2 x 2 + a 1 x + a 0 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