Module 5 Review 5.1 Cubic Functions

Algebra 2 **Module 5 Review** **Name:** Key **Date:** **Hour:** **5.1 - Cubic Functions** For 1-2, describe, in words, the transformations applied to the graph of f(x) = x^3 to produce the graph of g(x). Then graph g(x) with at least 3 specific points. 1) g(x) = (x-3)^3 + 2 Transformations: Right 3, Up 2 2) g(x) = -3(x+2)^3 - 2 Transformations: Reflect over x-axis, Vertical Stretch by 3, Left 2, Down 2 For 3-4, given the general equation f(x) = a(x-h)^3 + k, write the specific equation for the graph. 3) [Image of a cubic function graph] g(x) = -1/8(x-6)^3 + 2 4) [Image of a cubic function graph] g(x) = -1/2(x+2)^3 - 2 5) [Image of a cubic function graph] g(x) = 1/4(x-1)^3 + 1 **Algebra 2** **Module 5 Review** **Name:** Key **Date:** **Hour:** **5.1 - Cubic Functions** For 5-6, write the specific equation of the form \(g(x)=a(\frac{1}{b}(x-h))^{3}+k\) after the given transformations of the graph of \(f(x)=x^{3}\). 5) A reflection across the x-axis, followed by a translation 11 units up and 7 units to the left. \(g(x)=-(x+7)^{3}+11\) 6) A vertical stretch by a factor of 6, followed by a translation 9 units to the right and 3 units down. \(g(x)=6(x-9)^{3}-3\) **5.2 - Polynomial Functions** For 1-3, identify whether each function graphed has an odd or even degree and a positive or negative leading coefficient.   1) Degree: Even (at least 4th) Leading Coeff: + 2) Degree: Even (at least 4th) Leading Coeff: - 3) Degree: Odd (at least 5th) Leading Coeff: + For 4-5, graph each function on a graphing calculator to determine the number of turning points, the number of global maximum and/or   minimum values, and the number of local maximum and/or minimum values that are not global.   4) \(f(x)=x(x-4)^{2}\) Turning Points: 2 Global Maximum(s): 0 Local Maximum(s): 0 Global Minimum(s): 1 Local Minimum(s): 1 5) \(f(x)=-x^{2}(x-2)(x+1)\) Turning Points: 3 Global Maximum(s): 1 Local Maximum(s): 1 Global Minimum(s): 0 Local Minimum(s): 1 6) \(f(x) = -(x-1)^2(x+3)\) Degree: 3rd End Behavior: As \(x \rightarrow \infty, f(x) \rightarrow -\infty\); As \(x \rightarrow -\infty, f(x) \rightarrow \infty\) x-intercept(s): \(x = -3, 1\) y-intercept: \(y = -3\) Positive: \((-3, 1)\) Negative: \((-\infty, -3) \cup (1, \infty)\) 7) \(f(x) = (x+2)(x-2)(x-1)\) Degree: 3rd End Behavior: As \(x \rightarrow \infty, f(x) \rightarrow \infty\); As \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) x-intercept(s): \(x = -2, 1, 2\) y-intercept: \(y = 4\) Positive: \((-2, 1) \cup (2, \infty)\) Negative: \((-\infty, -2) \cup (1, 2)\) 8) \(f(x) = x(x-4)(x+1)\) Degree: 3rd End Behavior: As \(x \rightarrow \infty, f(x) \rightarrow \infty\); As \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) x-intercept(s): \(x = -1, 0, 4\) y-intercept: \(y = 0\) Positive: \((-1, 0) \cup (4, \infty)\) Negative: \((-\infty, -1) \cup (0, 4)\) 9) \(f(x) = -x^2(x+3)\) Degree: 3rd End Behavior: As \(x \rightarrow \infty, f(x) \rightarrow -\infty\); As \(x \rightarrow -\infty, f(x) \rightarrow \infty\) x-intercept(s): \(x = 0, -3\) y-intercept: \(y = 0\) Positive: \((-\infty, -3)\) Negative: \((0, \infty) \cup (-3, 0)\) 10) \(f(x) = (x+3)(x+1)(x)(x-2)\) 11) \(f(x) = -(x+4)(x+1)(x-1)\) 12) \(f(x) = (x+3)(x+1)(x-1)(x-4)\) 13) \(f(x) = -(x+2)(x-3)\) **For 12-13, write a function, V(x), for the volume of the box that could be made by cutting an x in. by x in. square from each corner of the given rectangular piece of cardboard. Then, graph your function to   answer the questions that follow. Approximate decimal answers to the nearest tenth.** 12) 8 in. x 15 in. \(V(x) = x(8-2x)(15-2x)\) A. What is the maximum possible volume? V = 90.7 in³ B. What value of x maximizes the volume? x = 1.7 in C. What are the dimensions of the box with the maximum volume? \(l = 4.6 in, w = 11.6 in, h = 1.7 in\) D. What are the domain and range of the function? D: \(0 \le x \le 4\) 13) 19 in. x 15 in. \(V(x) = x(15-2x)(19-2x)\) A. What is the maximum possible volume? V = 352.7 in³ B. What value of x maximizes the volume? x = 2.8 in C. What are the dimensions of the box with the maximum volume? \(l = 13.4 in, w = 9.4 in, h = 2.8 in\) D. What are the domain and range of the function? D: \(0 \le x \le 7.5\)