Investigating Graphs of Polynomial Functions How do we

Investigating Graphs of Polynomial Functions • How do we use properties of end behavior to analyze, describe, and graph polynomial functions? • How do we identify and use maxima and minima of polynomial functions to solve problems? Holt Mc. Dougal Algebra 2

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 Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions End behavior is a description of the values of the function as x approaches infinity (x +∞) or negative infinity (x –∞). The degree and leading coefficient of a polynomial function determine its end behavior. It is helpful when you are graphing a polynomial function to know about the end behavior of the function. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 1: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. Q(x) = –x 4 + 6 x 3 – x + 9 The leading coefficient is – 1, which is negative. The degree is 4, which is even. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 2: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. 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. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 3: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. P(x) = 2 x 4 + 3 x 2 – 4 x – 1 The leading coefficient is 2, which is positive. The degree is 4, which is even. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 4: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. S(x) = – 3 x 3 + x + 1 The leading coefficient is – 3, which is negative. The degree is 3, which is odd. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 5: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. P(x) has a negative leading coefficient and an odd degree. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 6: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. P(x) has a positive leading coefficient and an even degree. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 7: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. P(x) has a negative leading coefficient and an odd degree. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 8: Using Graphs to Analyze Polynomial Functions Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. P(x) has a positive leading coefficient and an even degree. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Now that you have studied factoring, solving polynomial equations, and end behavior, you can graph a polynomial function. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 9: Graphing Polynomial Functions Graph the function f(x) = x 3 + 4 x 2 + x – 6. Step 1 Find the roots of the equations. These are the x-intercepts. Holt Mc. Dougal Algebra 2 The y-intercept is -6.

Investigating Graphs of Polynomial Functions Example 9: Graphing Polynomial Functions Graph the function f(x) = x 3 + 4 x 2 + x – 6. Step 2 Determine end behavior of the graph. The leading coefficient is 1, which is positive. The degree is 3, which is odd. Because all of the zeros are multiplicity of 1, they go straight through the x-axis Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 10: Graphing Polynomial Functions Graph the function f(x) = x 3 – 2 x 2 – 5 x + 6. Step 1 Find the roots of the equations. These are the x-intercepts. Holt Mc. Dougal Algebra 2 The y-intercept is 6.

Investigating Graphs of Polynomial Functions Example 10: Graphing Polynomial Functions Graph the function f(x) = x 3 – 2 x 2 – 5 x + 6. Step 2 Determine end behavior of the graph. The leading coefficient is 1, which is positive. The degree is 3, which is odd. Because all of the zeros are multiplicity of 1, they go straight through the x-axis Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 11: Graphing Polynomial Functions Graph the function f(x) = – 2 x 2 – x + 6. Step 1 Find the roots of the equations. These are the x-intercepts. The y-intercept is 6. Holt Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Example 11: Graphing Polynomial Functions Graph the function f(x) = – 2 x 2 – x + 6. Step 2 Determine end behavior of the graph. The leading coefficient is -2, which is negative. The degree is 2, which is even. Because all of the zeros are multiplicity of 1, they go straight through the x-axis Holt Mc. Dougal Algebra 2

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 Mc. Dougal Algebra 2

Investigating Graphs of Polynomial Functions Lesson 4. 2 Practice A Holt Mc. Dougal Algebra 2