Algebra 2 Problems of the Day (open-ended)

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January 2018, Part III

Questions in Part II are worth 4 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.


35. In a random sample of 250 men in the United States, age 21 or older, 139 are married. The graph below simulated samples of 250 men, 200 times, assuming that 139 of the men are married.

a) Based on the simulation, create an interval in which the middle 95% of the number of married men may fall. Round your answer to the nearest integer.
b) A study claims “50 percent of men 21 and older in the United States are married.” Do your results from part a contradict this claim? Explain.

Answer:
The interval would be the mean minus twice the standard deviation to the mean plus twice the standard deviation.
138.905 - 2(7.950) to 138.905 + 2(7.950)
123.005 to 154.805
123 to 155 (please round correctly)

Fifty percent of the men in the survey would be 250 / 2 = 125, which falls between 123 and 155, so the claim is not contradicted by the results in part a because 123 < 125 < 155.



36. The graph of y = f(x) is shown below. The function has a leading coefficient of 1.

Write an equation for f(x).
The function g is formed by translating function f left 2 units. Write an equation for g(x).

Answer:
We have three zeroes: -4, 0, and 3. However, 0 is a double root because there's a local maximum on the x-axis. (The graph is obviously not cubic, as it goes up on both ends.)

An equation for this function is f(x) = (x + 4)(x²)(x - 3)
If you want to check, you can enter this into your graphing calculator, but be sure to adjust the scale. Notice that the y-axis has a scale of 10, while the x-axis has a scale of 1.

To find g, translate f by adding 2 to each of the factors.
g(x) = (x + 6)(x+2)²(x - 1)
Note: watch that you don't write "(x² + 2)" by accident when you're copying your f function!

Alternative answer: Suppose you got f(x) = (x + 4)(x²)(x - 3), but you thought you had to multiply it. That's okay, as long as you multiplied it correctly.
(x + 4)(x - 3) = x² -3x + 4x - 12 = x² + x - 12
(x² + x - 12)(x²) = x⁴ + x³ - 12x²

Using this approach, the easiest way to find g is to add the offset to each term, rather than adding 2 to each term and multiplying again!
g(x) = (x + 2)⁴ + (x + 2)³ - 12(x + 2)²



Comments and questions welcome.

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