SOLUTIONS. Math 110 Quiz 3 (Sections )

Transcription

1 SOLUTIONS Name: Section circle one): LSA 001) Engin 002) Math 110 Quiz 3 Sections ) Available: Wednesday, November 1 Friday, November 3 1. You have 30 minutes to complete this quiz. Keeping track of time is your responsibility. If you do not return this quiz after 30 minutes, your grade on this quiz may be lowered. 2. Show an appropriate amount of work including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. 3. Math Lab staff will not answer questions about the content of this quiz. 4. You may use a calculator on this quiz. 5. No other aids notes, textbooks, computers, phones, smartwatches, etc.) are allowed. 6. Turn off or silence all phones and other electronic devices. 7. Do not discuss the contents of this quiz with anyone other than a Math 110 instructor until Friday, November 3 at 4:00 PM. Problem Points Score Total: 29 Math Lab manager print and sign name): Start time stamp: End time stamp:

2 Math 110 Quiz 3 Fall 2017) page 2 1. [6 points] a. [3 points] Find a formula for the quadratic function p whose graph is shown below. 10 y y = px) Solution: px) has zeros at x = 4 and x = 8, so px) must have factored form px) = ax + 8)x 4) 8 4 x for some constant a. Plug in p0) = 10, 10 = a0 + 8)x 4) and solve to get a = 10/32. Answer: px) = 10 x + 8)x 4) 32 b. [3 points] Find a formula for the quadratic function r if the graph of y = rx) is a parabola with vertex 8, 15) and r10) = 27. Solution: rx) must have vertex form rx) = bx 8) for some constant b. Plug in r10) = 27, 27 = b10 8) and solve for b to get b = 3. Answer: rx) = 3x 8)

3 Math 110 Quiz 3 Fall 2017) page 3 2. [5 points] Baby Eloise has just learned to throw her toy ball in the air. While Eloise s mother is holding her, she throws the ball at time t = 0, and after she throws the ball, the equation of the ball s height above the ground, in meters, is given by ht) = 4.9t 2 + 8t a. [2 points] How high is the ball above the ground when Eloise releases the ball? Include units. Answer: 1.4 meters b. [3 points] After Eloise throws the ball, it continues in the air until it lands on a coffee table whose surface is 0.5 meters above the ground. How many seconds after Eloise throws the ball does it land on the coffee table? Show your work, then give your answer as a decimal accurate to three decimal places. Solution: We need to find when ht) = 0.5. We set up and subtract 0.5 to get Using the quadratic formula gives 0.5 = 4.9t 2 + 8t = 4.9t 2 + 8t t = 8) ± 8) )0.9) 2 4.9) or Since Eloise throws the ball at time t = 0, the negative value of t is not correct that would be before she even threw the ball). Answer: about seconds

4 Math 110 Quiz 3 Fall 2017) page 4 3. [4 points] In each part, circle if the statement is always and circle if the statement could be. You do not need to show work or explain your reasoning. a. [1 point] If s is a quadratic function such that the graph of y = sx) has vertex 0, 10) and such that s7) = 2, then it must be that s 7) = 2. Solution: If the vertex of the graph of y = sx) is at the point 0, 10), then the graph of y = sx) is a parabola that is symmetric about the line x = 0. By this symmetry, if s7) = 2, then s 7) must also be 2. b. [1 point] If f is a quadratic function with exactly one zero at x = k, then fx) = ax k) for some constant a. Solution: Note that fx) = ax k) is a linear function, not a quadratic function. A quadratic function with exactly one zero at x = k which is a quadratic function whose vertex is on the x-axis) will have the form fx) = ax k) 2 for some constant a. c. [1 point] The quadratic function Qx) = 3x + 4) is concave down for x < 4 and concave up for x > 4. Solution: The graph of a quadratic function is a parabola that opens up or a parabola that opens down, which means that a quadratic function is either concave up everywhere or concave down everywhere. The quadratic function Qx) has a positive leading coefficient, so it opens up and is concave up for all x. d. [1 point] If l is a linear function and t is a quadratic function, then the function F defined by F x) = tlx)) is a quadratic function. Solution: Full credit was given for both answers. If lx) is a nonconstant linear function, then lx) = px + r for some numbers p and r. If tx) is a quadratic function, then tx) = ax 2 + bx + c for some numbers a, b, and c. Then tlx)) = apx + r) 2 + bpx + r) + c = ap 2 )x 2 + 2apr + bp)x + ar 2 + br + c) is a quadratic function. However, a constant function is still considered to be a linear function. If lx) is a constant function, then tlx)) is also a constant function, and constant functions are not considered to be quadratic functions.

5 Math 110 Quiz 3 Fall 2017) page 5 4. [5 points] A company is selling candy corn by the pound. They start selling candy corn on October 1 and continue selling it until November 11. The price of the candy corn for the whole month of October is $4/lb. The price of the candy corn then decreases during November. The price of the candy corn for the first eleven days of November is given by a parabola whose vertex occurs on Halloween October 31), when the price is still $4/lb. On November 11, the price of the candy corn has reached $0/lb and they just give away the remaining candy corn. Write a formula for a piecewise-defined function cd) that models the price of candy corn, in dollars per pound, between October 1 and November 11 d days after September 30 so that d = 1 corresponds to October 1 and d = 42 corresponds to November 11). Note that part of your score on this problem will be for using correct mathematical notation for a piecewise function. Solution: Note that there was a typo in this problem: November 11 is d = 42, not d = 44. This solution is for the corrected version using d = 42. If you used d = 44, you will instead get 4 if 1 d 31 cd) = d 31)2 + 4 if 31 < d 44 It might help to think about the graph of cd), which is shown below. 4 cd) d cd), which is the price the company charges, is constant from Oct 1 through Oct 31 d = 1 through d = 31). On the time interval 31 t 42, cd) is a quadratic function with vertex 31, 4) and with a zero at d = 42. So for these values of t, cd) is a quadratic function that can be written in vertex form as cd) = ad 31) Plug in c42) = 0, giving and solve to get a = 4/ = a42 31) Answer: 4 if 1 d 31 cd) = d 31)2 + 4 if 31 < d 42

6 Math 110 Quiz 3 Fall 2017) page 6 5. [9 points] Consider the quadratic function fx) = 2x 2 + 5x 9. a. [5 points] Using the technique of completing the square, rewrite fx) in vertex form. Be sure to show all of your work. Solution: Important note: For full credit, you must use the method of completing the square to put this quadratic function in vertex form. On the midterm and final, you will receive little if any) credit for using a memorized shortcut to find the vertex. First factor out the leading coefficient 2: fx) = 2 x 2 52 ) x 9. Then we add and subtract 5/2)/2 ) 2 = 25/16: fx) = 2 x x ) Then, since we have x 5 ) 2 = x x , fx) = 2 x 5 ) ) Finally, we distribute the 2, giving fx) = 2 x 5 ) = 2 x 5 ) Answer: fx) = 2 x 5 ) b. [2 points] Based on your answer to part a, give the x-and y-coordinates of the vertex of the parabola y = fx). ) Answer: Vertex: 5/4, 47/8 c. [2 points] Is the vertex of f the maximum of the parabola, the minimum of the parabola, or neither? Circle one answer then briefly explain your reasoning. Answer: maximum minimum neither Explanation: Solution: The leading coefficient is negative, so the parabola is concave down it opens down ), making the vertex a maximum.