Math 4 quiz review October 27, 2016 Polynomial functions: review page 1 Quadratic and Polynomial functions: Quiz review

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1 October 27, 2016 Polynomial functions: review page 1 Quadratic and Polynomial functions: Quiz review Topic outline Quadratic functions Quadratic function formulas: you should be able to convert between forms including being able to put a function in vertex form using some method. o Standard form o Factored form o Vertex form Quadratic function graphs Polynomial functions Polynomial function formulas o standard form o factored form Polynomial function operations o factoring o finding zeros algebraically Polynomial function graphs o location of x-intercepts, based on factors and zeros o appearance of x-intercepts, based on multiplicity of factors o location of y-intercept, based on f(0) o end behavior, based on degree and leading coefficient Applications of polynomial functions (including quadratic applications). We will have a quiz on quadratics/polynomials on Friday 10/28 (H) or Monday 10/31 (A). You will be allowed to use your calculator on this test.

2 October 27, 2016 Polynomial functions: review page 2 Review Problems 1. In a rectangular piece of cardboard with perimeter 30 inches, two parallel and equally spaced creases are made, as shown below. The cardboard is then folded to make a prism with open ends that are equilateral triangles. Use this fact to help you solve problem 1: The area of an equilateral triangle where each side is x units long is: Area = 3 4 x2. a. Find a formula for the volume of the prism, V(x). b. What is the domain of V(x)? (i.e. what values of x make sense in this problem?) c. Find the approximate value of x that maximizes the volume. Then give the approximate maximum volume. 2. A pumpkin is dropped to the ground from the roof of a tall building. In 3.2 seconds, it hits the ground. Approximately how high was the roof? Give your answer in feet. 3. Let f(x) = 3x 2 36x a. Using any method, identify the vertex of f(x). b. Write f(x) in vertex form. 4. Write a possible function formula for this graph, using what you know about factors and multiplicity. (Note that there is not enough information to find the a value here, but you can tell whether a is positive or negative, then make a be any number of that kind.)

3 October 27, 2016 Polynomial functions: review page 3 5. Answer the following questions about the function f(x) = 2x 4 8x 3 + 6x 2. a. What are the degree and the leading coefficient of f(x)? Based on these, what do you know about the end behavior of the graph of f(x)? b. Factor f(x). Hint: First factor out a common factor, then factor further. c. List the x-intercepts of f(x). At each, tell whether the graph passes through or just touches the x- axis. d. With the information from parts a c, but without a calculator, sketch a possible graph of f(x). e. Graph f(x) on your calculator. Set the window to match your sketch as closely as possible. Record your window and graph.

4 October 27, 2016 Polynomial functions: review page 4 6. Sketch a graph of this function. Label the x-intercept(s), y-intercept, and any special points (such as a vertex) with coordinates. Your graph shape will be close enough if it has correct end-behavior and correct appearance (touch vs. pass-through) at each x-intercept. f(x) = 2 x (x + 1) 3 (x 2) 2 (4x 15) 7. Find a function formula for this polynomial graph. All of the zeros are visible on the graph, and there are no multiplicities higher than 2. (The only kinds of factors are (x r) s and (x r) 2 s.)

5 October 27, 2016 Polynomial functions: review page 5 Answers 1. a. V ( x) 3 = x 2 ( 15 3x) 4 b. 0 < x < 5 (why?) c. Max volume in 3 when x 3.33 in. 2. About 164 feet. 3. a. Vertex: (6, 8). b. f ( x) = 3( x 6) f(x) = (x + 3)(x + 1) 3 (x 2) 2 [other answers possible] 5. a. degree 4 (even), leading coefficient 2 (positive), so end behavior is up on both sides. b. 2x 2 (x 1)(x 3) or x 2 (2x 2)(x 3) or x 2 (x 1)(2x 6) c. 0 (just touches), 1 (passes through), 3 (passes through) d and e can be checked by making sure that they agree with each other. 6. Self-check the shape of the graph by graphing the function on your calculator. You might need to adjust the window to get a view matching what you drew (especially, try larger Ymin and Ymax). Make sure that you have these features labeled with numbers: x-intercept just touching at 2; x-intercepts passing through at 0, 1, and f(x) = 1 2 ( x + 4)( x 3) ( x 6) 96 x (any value of a close to 1/100 is correct.)