2-2: Evaluate and Graph Polynomial Functions
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1 2-2: Evaluate and Graph Polynomial Functions What is a polynomial? -A monomial or sum of monomials with whole number exponents. Degree of a polynomial: - The highest exponent of the polynomial How do we write polynomials? -Standard form: terms in descending order of exponents from left to right. P(x) = 4x 3 + 3x 2-6x + 7 cubic term quadratic term linear term constant
2 Example of a polynomial: f (x) = 4x 4 + 2x 3 - x + 7 What is the constant? What is the degree? Classification of a polynomial by degree: Degree Name Example 0 constant 7 1 linear ½x-3 2 quadratic 8x cubic 2x 3-4x 2 +x-8 4 quartic 2x 4 + 3x 3-5x-11 5 quintic x 5-7x 4 +x 3-4x 2 +x-8
3 Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree and classification. a. f (x) = 2x 2 - x -2 b. f (x) = -0.8x 3 + x 4-5 Decide whether the function is a polynomial function. If it is, write the function in standard form, classify the polynomial by degree and by the number of terms. a. f (x) = x + 5x 3 b. f (x) = 6-4x + πx 4
4 Try these: Classify each polynomial by degree and number of terms. a) x x - 2x 3 b) 3x x 3-6x 5 Evaluating Functions: -To evaluate, just substitute the number into the function. Evaluate f (x) = 4x 4 + 2x 3 - x + 7 when x = -2
5 Polynomial Function Graphs Exploring End Behavior What is end behavior? -the behavior of the graph positive or negative infinity. Graph: y = 2x 2-3x + 4 leading coefficient degree From the Left: From the Right:
6 Summary: If the leading coefficient is positive, the graph rises to the right. If the leading coefficient is negative, the graph falls to the right. When the function's degree is odd, the ends will go in opposite directions. When the function's degree is even the ends go in the same direction.
7 Describe the end behavior of the graph. Explain your answer. f (x) = x 5 + 2x 2 - x + 4 f (x) = 4x 4 + 2x 3 - x + 7 Describe the end behavior of the graph. Explain your answer. f (x) = -x 3 + 3x 2 + 6x - 2 f (x) = -x 6-4x 3 + 8x 2 + 7
8 local maximum - the highest point of the function in that area of the graph Turning Points local minimum - the lowest point of the function in that area of the graph maximum minimum *A cubic function has at most 2 turning points. *A quartic function has at most 3 turning points, and so on. f(x) = x 3-3x x-intercepts: Turning Points: Increasing: Decreasing:
9 f(x) = x 4-6x 3 + 3x x - 3 x-intercepts: Turning Points: Increasing: Decreasing: f(x) = x 4 + 3x 3 - x 2-4x - 5 x-intercepts: Turning Points: Increasing: Decreasing:
10 You are making a rectangular box out of a 16-inch by 20- inch piece of cardboard. The box will be formed by making the cuts shown in the diagram and folding up the sides. You want to make the box to have the greatest volume possible. How long should you make the cuts? What is the maximum volume? What will the dimensions of the finished box be? Assignment: p. 99 # 4, 6,12 p. 100 # even (no calculator) p. 148 # 22 (calculator)
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