Analyze graphs of Polynomial Functions Lesson 2.8 Honors Algebra 2 Analyze graphs of Polynomial Functions Lesson 2.8

Goals Goal Rubric Determine the end behavior of a polynomial function. Find the number turning points of a polynomial function. Use intercepts to graph polynomial function. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Local Maximum Local Minimum

End Behavior 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.

End Behavior End behavior is a description of the values of the function as x approaches infinity (x → +∞), the right side of the graph or negative infinity (x → –∞), the left side of the graph. 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.

End Behavior

Identify the leading coefficient, degree, and end behavior. Example: Determining End Behavior of Polynomial Functions Identify the leading coefficient, degree, and end behavior. A. Q(x) = –x4 + 6x3 – 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) = 2x5 + 6x4 – x + 4 The leading coefficient is 2, which is positive. The degree is 5, which is odd. As x → –∞, P(x) → –∞, and as x → +∞, P(x) → +∞.

Identify the leading coefficient, degree, and end behavior. Your Turn: Identify the leading coefficient, degree, and end behavior. a. P(x) = 2x5 + 3x2 – 4x – 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) = –3x2 + x + 1 The leading coefficient is –3, which is negative. The degree is 2, which is even. As x → –∞, P(x) → –∞, and as x → +∞, P(x) → –∞.

Example: 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 odd degree with a negative leading coefficient.

Example: 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 even degree with a positive leading coefficient.

Your Turn: 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 odd degree with a negative leading coefficient.

Your Turn: 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 even degree with a positive leading coefficient.

Graph the function f (x) = (x + 3)(x – 2)2. EXAMPLE 1 Use x-intercepts to graph a polynomial function Graph the function f (x) = (x + 3)(x – 2)2. 1 6 SOLUTION STEP 1 Plot: the intercepts. Because –3 and 2 are zeros of f, plot (–3, 0) and (2, 0). STEP 2 Plot: points between and beyond the x-intercepts.

EXAMPLE 1 Use x-intercepts to graph a polynomial function STEP 3 Determine: end behavior. Because f has three factors of the form x – k and a constant factor of , it is a cubic function with a positive leading coefficient. So, f (x) → – ∞ as x → – ∞ and f (x) → + ∞ as x → + ∞. 1 6 STEP 4 Draw the graph so that it passes through the plotted points and has the appropriate end behavior.

Turning Point 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.

Number of Turning Points A linear function has degree 1 and no turning points. A quadratic function has degree 2 with one turning point. A cubic function has degree 3 with at most two turning points. A quartic function has degree 4 with at most three turning points.

Turning Points Extending this idea: Number of Turning Points The number of turning points of the graph of a polynomial function of degree n  1 is at most n – 1. A polynomial function of degree n has at most n – 1 turning points and at most n x-intercepts. If the function has n distinct roots, then it has exactly n – 1 turning points and exactly n x-intercepts. You can use a graphing calculator to graph and estimate maximum and minimum values.

EXAMPLE 2 Find turning points Graph the function, using the graphing calculator. Identify the x-intercepts and the points where the local maximums and local minimums occur. a. f (x) = x3 – 3x2 + 6 b. g (x) = x4 – 6x3 + 3x2 + 10x – 3

Use a graphing calculator to graph the function. EXAMPLE 2 Find turning points SOLUTION a. f (x) = x3 – 3x2 + 6 Use a graphing calculator to graph the function. Notice that the graph of f has one x-intercept and two turning points. You can use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points. The x-intercept of the graph is x ≈ –1.20. The function has a local maximum at (0, 6) and a local minimum at (2, 2). ANSWER

Use a graphing calculator to graph the function. EXAMPLE 2 Find turning points SOLUTION b. g (x) = x4 – 6x3 + 3x2 + 10x – 3 Use a graphing calculator to graph the function. Notice that the graph of g has four x-intercepts and three turning points. You can use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points. The x-intercepts are x ≈ –1.14, x ≈ 0.29, x ≈ 1.82, and x ≈ 5.03. The function has a local maximum at (1.11, 5.11) and local minimums at (–0.57, –6.51) and (3.96, – 43.04). ANSWER

Your Turn: for Examples 1, 2 and 3 Graph the function, using the graphing calculator. Identify the x-intercepts and the points where the local maximums and local minimums occur. 1. f(x) = 0.25(x + 2)(x – 1)(x – 3) ANSWER x-intercepts: –2, 1, 3 local minimum: (2.1, –1.0) local maximum: (–0.79, 2.1)

x-intercepts: 1, 4 local minimum: (3, –8) local maximum: (–1, 0) Your Turn: for Examples 1, 2 and 3 Graph the function, using the graphing calculator. Identify the x-intercepts and the points where the local maximums and local minimums occur. 2. g (x) =2(x – 1)2(x – 4) ANSWER x-intercepts: 1, 4 local minimum: (3, –8) local maximum: (–1, 0)

Your Turn: for Examples 1, 2 and 3 Graph the function, using the graphing calculator. Identify the x-intercepts and the points where the local maximums and local minimums occur. 3. h (x) = 0.5x3 + x2 – x + 2 ANSWER x-intercept: –3.1 local minimum: (0.39, 1.8) local maximum: (–1.7, 4.1)

Your Turn: for Examples 1, 2 and 3 Graph the function using the graphing calculator. Identify the x-intercepts and the points where the local maximums and local minimums occur. 4. f (x) = x4 + 3x3 – x2 – 4x – 5 ANSWER x-intercepts: –3.1, 1.4 local minimums: (–2.3, –9.6), (0.68, –7.0) local maximum: (–0.65, –3.5)

• How long should you make the cuts? EXAMPLE 3 Maximize a polynomial model 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 the box to have the greatest volume possible. Arts And Crafts • How long should you make the cuts? • What is the maximum volume? • What will the dimensions of the finished box be?

Write a verbal model for the volume. Then write a function. EXAMPLE 3 Maximize a polynomial model SOLUTION Write a verbal model for the volume. Then write a function.

To find the maximum volume, graph the volume EXAMPLE 3 Maximize a polynomial model = (320 – 72x + 4x2)x Multiply binomials. = 4x3 – 72x2 + 320x Write in standard form. To find the maximum volume, graph the volume function on a graphing calculator. Consider only the interval 0 < x < 8 because this describes the physical restrictions on the size of the flaps.

From the graph, you can see that the maximum EXAMPLE 3 Maximize a polynomial model From the graph, you can see that the maximum volume is about 420 and occurs when x ≈ 2.94. You should make the cuts about 3 inches long.The maximum volume is about 420 cubic inches. The dimensions of the box with this volume will be about 3 inches by 10 inches by 14 inches. ANSWER

5. WHAT IF? In Example 3, how do the answers Your Turn: for Examples 1, 2 and 3 5. WHAT IF? In Example 3, how do the answers change if the piece of cardboard is 10 inches by 15 inches? ANSWER The cuts should be about 2 inches long. The maximum volume is about 132 cubic inches. The dimensions of the box would be 6 inches by 11 inches by 2 inches.