Investigating Graphs of of Investigating Graphs 6 7
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Investigating Graphs of of Investigating Graphs 6 -7 Polynomial Functions Objectives Use properties of end behavior to analyze, describe, and graph polynomial functions. Identify and use maxima and minima of polynomial functions to solve problems. Holt Algebra 22
6 -7 Investigating Graphs of Polynomial Functions Polynomial functions are classified by their degree. The graphs of polynomial functions are classified by the degree of the polynomial. Each graph, based on the degree, has a distinctive shape and characteristics. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions End behavior is a description of the values of the function as x approaches infinity (x +∞) goes to the right or negative infinity (x –∞) The degree and leading coefficient of a polynomial function determine its end behavior. It is helpful when you are goes to the left. graphing a polynomial function to know about the end behavior of the function. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions END BEHAVIOR – be the polynomial • The LEADING COEFFICIENT is either positive or negative • Positive, put your right arm up; negative, put your right arm down • The DEGREE is either even or odd • Even, arms together; odd, arms apart (or opposite) Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Determine the end behavior: 1. 4 x 4 – 2 x 3 + 6 x – 3 = 0 Leading Coefficient → POSITIVE → right arm up Degree → EVEN → arms together Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Determine the end behavior: 2. 3 x 7 + 8 x 2 + 4 x – 13 = 0 Leading Coefficient → POSITIVE → right arm up Degree → ODD → arms apart Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Determine the end behavior: 3. -2 x 5 + x 4 - 6 x 2 – 8 x = 0 Leading Coefficient → NEGATIVE → right arm down Degree → ODD → arms apart Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Determine the end behavior: 4. -2 x 2 – 6 x + 6 = 0 Leading Coefficient → NEGATIVE → right arm down Degree → EVEN → arms together Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Example 1: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. A. Q(x) = –x 4 + 6 x 3 – x + 9 The leading coefficient is – 1, which is negative. The degree is 4, which is even. As x +∞, P(x) –∞, and as x – ∞, P(x) –∞. B. P(x) = 2 x 5 + 6 x 4 – x + 4 The leading coefficient is 2, which is positive. The degree is 5, which is odd. As x +∞, P(x) +∞, and as x Holt Algebra 2 -∞, P(x) -∞.
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 1 Identify the leading coefficient, degree, and end behavior. a. P(x) = 2 x 5 + 3 x 2 – 4 x – 1 The leading coefficient is 2, which is positive. The degree is 5, which is odd. As x +∞, P(x) +∞, and as x -∞, P(x) -∞. b. S(x) = – 3 x 2 + x + 1 The leading coefficient is – 3, which is negative. The degree is 2, which is even. As x +∞, P(x) –∞, and as x Holt Algebra 2 -∞, P(x) –∞.
6 -7 Investigating Graphs of Polynomial Functions Example 2 A: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. As x +∞, P(x) -∞, and as x -∞, P(x) +∞. P(x) is of negative leading coefficient with an odd degree. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Example 2 B: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. Right arm is up, LC was positive, Arms together, Degree was even As x +∞, P(x) +∞, and as x -∞, P(x) +∞. P(x) is of positive leading coefficient with an even degree. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 2 a Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. Right arm is down, LC was negative, Arms apart, Degree was odd As x +∞, P(x) -∞, and as x -∞, P(x) +∞. P(x) is of negative leading coefficient with an odd degree. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 2 b Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. Right arm is up, LC was positive, Arms together, Degree was even As x –∞, P(x) +∞, and as x +∞, P(x) +∞. P(x) is of even degree with a positive leading coefficient. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Homework Page 457 (2 -9, 15 -22) Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Now that you have studied factoring, solving polynomial equations, and end behavior, you can graph a polynomial function. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Example 3: Graphing Polynomial Functions Graph the function. f(x) = x 3 + 4 x 2 + x – 6. ZEROS: Y – intercept: End Behavior: Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 3 a Graph the function. f(x) = x 3 – 2 x 2 – 5 x + 6. ZEROS: Y – intercept: End Behavior: Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 3 b Graph the function. f(x) = – 2 x 2 – x + 6. ZEROS: Y – intercept: End Behavior: Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions A turning point is where a graph changes from increasing to decreasing or from decreasing to increasing. A turning point corresponds to a local maximum or minimum. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Example 4: Determine Maxima and Minima with a Calculator Graph f(x) = 2 x 3 – 18 x + 1 on a calculator, and estimate the local maxima and minima. Step 1 Graph. 25 The graph appears to have one – 5 local maxima and one local minima. Step 2 Find the maximum. Press to access the CALC menu. Choose 4: maximum. The local maximum is approximately 21. 7846. Holt Algebra 2 5 – 25
6 -7 Investigating Graphs of Polynomial Functions Example 4 Continued Graph f(x) = 2 x 3 – 18 x + 1 on a calculator, and estimate the local maxima and minima. Step 3 Find the minimum. Press to access the CALC menu. Choose 3: minimum. The local minimum is approximately – 19. 7846. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 4 a Graph g(x) = x 3 – 2 x – 3 on a calculator, and estimate the local maxima and minima. Step 1 Graph. 5 The graph appears to have one – 5 local maxima and one local minima. Step 2 Find the maximum. Press to access the CALC menu. Choose 4: maximum. The local maximum is approximately – 1. 9113. Holt Algebra 2 5 – 5
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 4 a Continued Graph g(x) = x 3 – 2 x – 3 on a calculator, and estimate the local maxima and minima. Step 3 Find the minimum. Press to access the CALC menu. Choose 3: minimum. The local minimum is approximately – 4. 0887. Holt Algebra 2
6 -7 Investigating Graphs of Polynomial Functions Check It Out! Example 4 b Graph h(x) = x 4 + 4 x 2 – 6 on a calculator, and estimate the local maxima and minima. 10 Step 1 Graph. The graph appears to have one local maxima and one local minima. – 10 Step 2 There appears to be no maximum. Step 3 Find the minimum. Press to access the CALC menu. Choose 3: minimum. The local minimum is – 6. Holt Algebra 2 10 – 10
6 -7 Investigating Graphs of Polynomial Functions HOMEWORK: Holt Algebra 2
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