This exam was adminstered in June 2024 .
June 2024 Algebra 2, Part II
Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.
25. Given x is a real number, write the expression in simplest a + bi form:
(x + 2i)(3 - 2xi) + 2x2i
Answer:
Multiply the binomials using FOIL or the Box Method, keeping in mind that i2 = -1. Then combine like terms.
(x + 2i)(3 - 2xi) + 2x2i
3x - 2x2i + 6i - 4xi2 + 2x2i
3x - 2x2i + 6i + 4x + 2x2i
3x + 6i + 4x
7x + 6i
26. Solve 3.8e1.5t = 16 algebraically for t to the nearest hundredth.
Answer:
Inverse operations. Have a calculator handy.
3.8e1.5t = 16
e1.5t = 16/3.8
ln (e1.5t) = ln(16/3.8)
1.5t = 1.43758765551
t = 0.95839177033
t = 0.96 to the nearest hundredth.
27. In an attempt to get the student body’s opinion of a new dress code, members of the statistics class
surveyed the students of the first period computer science class. Explain a statistical bias in the
method of data collection.
Answer:
There are several possible answers that would show bias.
The computer science class is not representative of how the entire student body dresses.
Only surveying first period students excludes students who have schedules that start later.
Other answers are possible, but basically, point out that this is not a random sample of students in the school.
28. Sketch a graph of polynomial P(x), given the criteria below:
• P(x) has zeros only at -5, 1, and 4
• As x → ∞, P(x) → -∞
• As x → -∞, P(x) → -∞
Answer:
The graph must start at the bottom and end at the bottom because it comes up from negative infinity and goes off to negative infinity. In between it has to cross or touch the x-axis at -5, 1, and 4. One of those must be a double root. That is, the curve will touch the x-axis and turn back instead of crossing the axis.
A possible solution looks like this:
It's just a sketch, so it doesn't have to be perfectly measured out. However, if you draw tick marks on the x-axis, then the zeros better be at -5, 1, and 4. You should have arrows on the ends of the lines to indicate that the graph continues in the direction shown. (Seriously, you'll lose a point without them because that would suggest endpoints to the graph.)
The turnaround point could happen at any of the three points but it has to be at one of them. It could also be at all three of them if it was a sixth-degree polynomial, P(x) = (x+5)2(x-1)2(x-4)2.
29. The height, above ground, of a Ferris wheel car can be modeled by the function
h(t) = - 103.5cos(2πt/5) + 108.5 where h is measured in feet and t is measured in minutes.
State the period of the function and describe what the period represents in this context.
Answer:
The period of cos(t) is 2π. The period of cos (2πt/5) is 2π / (2π/5), which is 5. The period is 5.
In this context, the period is how long it takes the Ferris wheel to complete one full revolution, which in this case is 5 minutes.
That sounds like a lot but it's a really big Ferris wheel with a radius of 103.5 feet. The center is 108.5 feet above the ground.
30. Solve algebraically for all values of x:
8 / (x + 5) - 3 / x = 5
Answer:
Multiply each term by (x) and (x + 5) to eliminate the denominators. This will result in a quadratic equation that needs to be solved.
8 / (x + 5) - 3 / x = 5
8(x)(x + 5) / (x + 5) - (3 (x)(x + 5)) / x = 5(x)(x + 5)
8x - 3x - 15 = 5x2 + 25x
8x - 3x - 15 = 5x2 + 25x
0 = 5x2 + 20x + 15
0 = x2 + 4x + 3
0 = (x + 3)(x + 1)
x + 3 = 0 or x + 1 = 0
x = -3 or x = -1
Neither solution causes a zero denominator, so both are accepted and neither is discarded.
31. The transportation methods used by the upperclassmen at Calhoun High School are summarized
in the table below.
Are the events “being a junior” and “driving to school” independent? Using statistical evidence,
justify your answer.
Answer:
If "being a junior" and "driving to school" are independent then the probability of being a junior who drives would equalt the probability of being a junior times the probability of driving.
If you add up the subtotals, you will see that there are 145 juniors and 139 upperclassmen who drive. There are a total of 277 upperclassmen.
P(J and D) = 58/277 = .209
P(J) * P(D) = 145/277 * 139/277 = .263
The probabilites are not the same. Therefore "being a junior" and "driving to school" are NOT independent events.
You must have a concluding statement. You can just write the two decimals without explainging what they mean.
32. Can f(x) = x3 + 7 be classified as an odd function? Justify your answer
Answer:
No, f(x) is neither an odd or an even function.
An odd function will only have odd exponents on the variables. However, the constant term (x0 term) has an exponent of zero, which is even.
Odd functions have rotational symmetry. That means that odd functions must go through the origin. However, f(0) = 7, not zero, so it can't be an odd function.
Finally, for a function to be odd f(-x) must equal -f(x). So f(-1) must equal -f(1) and f(0) must equal -f(0).
F(-1) = 6, but -f(1) = -8, and f(-0) = 7, but -f(0) = -7. So f(x) is not odd.
In general, f(-x) = (-x)3 + 7, but -f(x) = -(x3 + 7) = -(x3) - 7. so f(-x) =/= -f(x).
Either of the two explanations or the examples above are sufficient for full credit.
End of Part II
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