August 2018 Algebra I, Part I
Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.
1. The number of bacteria grown in a lab can be modeled by
P(t) = 300 • 24t, where t is the number of hours. Which expression is equivalent to P(t)?
Answer: (2) 300 • 16t
24 = 16, so 24t = so (24)4t = 16t
2. During physical education class, Andrew recorded the exercise times
in minutes and heart rates in beats per minute (bpm) of four of his classmates. Which table best represents a linear model of exercise
time and heart rate?
Answer: (1) [IMAGE OMITTED]
Table 1 shows a linear model. There is a constant rate of change of 5 bpm per 1 minute.
3. David correctly factored the expression m2 - 12m - 64.
Which expression did he write?
Answer: (3) (m - 16)(m + 4)
If you graph the original expression, you would see the zeroes are at -4 and 16. Flip the signs for the factored form.
If you multiply the choices, you will see that Choice (1) has a middle tern of -16m and (2) has a middle term of 0.
The factors of -64 that add up to -12 are -16 and +4.
4. The solution to -2(1 - 4x) = 3x + 8 is
Answer: (2) 2
Again, you could graph both sides of the equation and find the point of intersection.
And you can work backward from the choices (if you don't have a problem with fractions).
-2(1 - 4x) = 3x + 8
-2 + 8x = 3x + 8
8x = 3x + 10
5x = 10
x = 2
Check: -2(1 - 4(2)) = 3(2) + 8?
-2(-7) = 6 + 8
14 = 14
5. The graph of f(x) is shown below.
What is the value of f(3)?
Answer: (1) 6
Look at the graph. When x = -3, f(x) = 6.
6. If the function f(x) = x2 has the domain {0, 1, 4, 9}, what is its range?
Answer: (2) {0, 1, 16, 81}
The domain is the set of x values (the input to the function), and the range is the set of y values (the output of the function.)
The output is the square of the input. And squared numbers are always positive, so (4) makes no sense.
(1) shows the positive square root function. (3) shows both positive and negative and would NOT be a function.
7. The expression 4x2 - 25 is equivalent to
Answer: (3) (2x + 5)(2x - 5)
Difference of Squares rule. 4x2 is a perfect square. 25 is a perfect square.
Break it into two conjugates with the square root of each term. (Conjugates are the same two terms but one has a plus and one has a minus. When you multiply them, the inner and outer terms will "cancel".)
8. Compared to the graph of f(x) = x2, the graph of g(x) = (x - 2)2 + 3
is the result of translating f(x)
Answer: (3) 2 units right and 3 units up
First of all, this should have been a "gimme" because "2 units left and 3 units up" wasn't a choice.
The +3 shifts the graph UP, which eliminates (1) and (4).
The -2 shifts the graph to the RIGHT, which is (3). "Down" in (2) is just silly.
9. Lizzy has 30 coins that total $4.80. All of her coins are dimes, D, and
quarters, Q. Which system of equations models this situation?
Answer: (2) D + Q = 30 ; .10D + .25Q = 4.80
The sum of the coins is 30. The value of those coins .10D + .25Q
10. Gretchen has $50 that she can spend at the fair. Ride tickets cost $1.25 each and game tickets cost $2 each. She wants to go on a minimum of
10 rides and play at least 12 games.
Which system of inequalities represents this situation when r is the
number of ride tickets purchased and g is the number of game tickets
purchased?
Answer: (2) 1.25r + 2g < 50, r > 10, g > 12
The most see can spend if $50, so less than or equal to. Eliminate (1) and (4).
The minimum numbers are 10 rides and 12 games, so greater than or equal to.
This is a "silly" question based on the choices. The third sentence pretty much tells you that the two inequalities involving r and g have to have the same symbol. They are only the same in one of the four choices.
11. Three functions are shown
[IMAGE TO BE ADDED]
Which statement is true?
Answer: (4) The y-intercept for g(x) is greater than the y-intercept for both f(x)
and h(x).
The y-intercept for f(x) is 1 (from the graph).
The y-intercept for g(x) is g(0) = 30 + 2 = 1 + 2 = 3.
The y-intercept for h(x) is -1 (from the table).
Even if you thought 30 was 0, and g(0) was 2, you still should have gotten this correct.
Note that choice (3) could not be correct without choice (1) also being correct, so (3) could be eliminated.
12. Olivia entered a baking contest. As part of the contest, she needs to demonstrate how to measure a gallon of milk if she only has a teaspoon measure. She converts the measurement using the ratios below:
4 quarts/1 gallon * 2 pints/1 quart * 2 cups/1 pint * 1/4 cup/4 tablespoons * 3 teaspoons/1 tablespoon
Which ratio is incorrectly written in Olivia's conversion?
Answer: (3) 1/4 cup/4 tablespoons
Short answer: in each case, the larger unit is on the bottom, except for this ratio.
Longer answer: to find the number of teaspoons in a gallon, all the in-between units need to cancel out, meaning that they will appear once in the numerator and once in the denominator. After pints are converted to cups, cups need to be converted to tablespoons. For this to happen, you need to know the number of tablespoons per cup, not cups per tablespoon.
13. If y = 3x3 + x2 - 5 and z = x2 - 12, which polynomial is equivalent to 2(y + z)?
Answer: (1) 6x3 + 4x2 - 34
3x3 + x2 - 5 + x2 - 12 = 3x3 + 2x2 - 17.
Multiply that sum by 2 and you get: 6x3 + 4x2 - 34
14. An outdoor club conducted a survey of its members. The members were asked to state their preference between skiing and snowboarding.
Each member had to pick one. Of the 60 males, 45 stated they preferred to snowboard. Twenty-two of the 60 females preferred to ski. What is the relative frequency that a male prefers to ski?
Answer: (2) 0.25
45 out of 60 males preferred to snowboard, so 15 out of 60 preferred to ski. (They had to choose one of the two.)
15/60 = 0.25.
15. When the function g(x) = {5x, x < 3; x2 + 4, x > 3} is graphed correctly, how should the points be drawn on the graph for an x-value of 3?
Answer: (4) a closed circle at (3,15) and an open circle at (3,13)
When x = 3, the first condition is used, so there is a closed circle at 5(3) = 15.
The second condition has a boundary for x = 3, so there is an open circle for (3)2 + 4, which is 9 + 4 = 13.
16. If f(x) = 2x2 + x - 3, which equation can be used to determine the zeros of the function?
Answer: (2) 0 = (2x + 3)(x - 1)
Factor the function. Choices (3) and (4) cannot be used because they contain subtraction, not just factors.
Alternatively, you can graph the original function, along with each choice, to see which gives you the same graph.
I. x represents the number of dogs at the shelter per day.
II. x represents the number of volunteers at the shelter per day.
III. 30 represents the shelter’s total expenses per day.
IV. 30 represents the shelter’s nonfood expenses per day.
17. Each day, a local dog shelter spends an average of $2.40 on food
per dog. The manager estimates the shelter’s daily expenses, assuming there is at least one dog in the shelter, using the function E(x) = 30 + 2.40x.
Which statements regarding the function E(x) are correct?
Answer: (2) I and IV
The independent variable is the number of dogs. The number of volunteers doesn't matter in this function, and there is no information about them. E(x) is the daily expenses, not 30, which is the nonfood expenses, other than the $2.40 spent per dog.
So statements I and IV are correct.
18. Which point is not in the solution set of the equation 3y + 2 = x2 - 5x + 17?
Answer: (1) (2,10)
Zeroes of a function. Factored form.
The shortcut for the graphing calculator is to put the following into y1
y1 = (x2 - 5x + 17 - 2) / 3
You don't need to be neat about it.
Check the table of contents.
Otherwise, plug in the values and see what makes a true statement:
(-2, 10): 3(10) + 2 = (-2)2 - 5(-2) + 17
32 + 2 = 4 + 10 + 17
34 =/= 31, Choice (1) is not a solution.
Check the others:
(-1, 7): 3(7) + 2 = (-1)2 - 5(-1) + 17
21 + 2 = 1 + 5 + 17
23 = 23, solution
(2, 3): 3(3) + 2 = (2)2 - 5(2) + 17
9 + 2 = 4 - 10 + 17
11 = 11, solution
(5, 5): 3(5) + 2 = (5)2 - 5(5) + 17
15 + 2 = 25 - 25 + 17
17 = 17, solution
19. The functions f(x) and g(x) are graphed below.
[IMAGE OMITTED]
Based on the graph, the solutions to the equation f(x) = g(x) are
Answer: (3) the x-values of the points of intersection
In any system of equations, you are looking for the points of intersection. In particular, with two functions, you are looking for the values of x that make the two functions (the y values) equal.
20. For the sequence -27, -12, 3, 18, …, the expression that defines the nth term where a1 = 27 is
Answer: (4) -27 + 15(n - 1)
The initial term is -27. The common difference (18 - 3 or 3 - (-12)) is 15, which is obvious from the choices. The common difference gets the variable, so choices (1) and (2) are eliminated.
When n = 1, an = a1 = -27
However, if you add 15(1) to -27, you get -12, not -27, so choice (3) is no good.
In choice (4), by subtracting 1 from 1, you multiply the 15 by 0, and the term goes away, leaving only the -27, which is what we want.
21. The data obtained from a random sample of track athletes showed that as the foot size of the athlete decreased, the average running speed
decreased. Which statement is best supported by the data?
Answer: (3) The sample of track athletes shows a correlation between foot size
and running speed.
There is a correlation between the two -- we can see that there is some relationship.
However, we don't know that one thing causes the other, so you can't say that there is a causal relationship.
Final note: Look at the word again. It does not say "casual" relationship, which is something two people might be engaged in. (Yes, I bring this up because people read it incorrectly the first time.)
22. Which system of equations will yield the same solution as the system below?
x - y = 3
2x - 3y = -1
Answer: (3) 2x - 2y = 6; 2x - 3y = -1
In Choice (3), the first equation was multiplied by 2 while the second was left alone.
In Choice (1), the first equation was multiplied incorrectly by -2. The second term should have been + 2y.
In Choice (2), only the left side of the first equation was multiplied by -2; the right side was not.
In Choice (4), the first equation was multiply incorrectly by 3. The second term should be - 3y.
23. Which of the three situations given below is best modeled by an exponential function?
I. A bacteria culture doubles in size every day.
II. A plant grows by 1 inch every 4 days.
III. The population of a town declines by 5% every 3 years.
Answer: (4) I and III
In Situation I, the bacteria doubles, then doubles again, etc. This is exponential.
In Situation II, the plant grows by the same amount, a constant rate. This is linear.
In Situation III, the population decreases by 5% then that decreases by 5%, etc. This is exponential.
24. The length, width, and height of a rectangular box are represented by 2x, 3x + 1, and 5x - 6, respectively. When the volume is expressed
as a polynomial in standard form, what is the coefficient of the 2nd term?
Answer: (3) -26
Volume = length X width X height
(2x)(3x + 1) = 6x2 + 2x
(6x2 + 2x)(5x - 6) = 30x3 - 36x2 + 10x2 - 12x
= 30x3 - 26x2 - 12x
The coefficient of the middle term is 26.
End of Part I
How did you do?
Questions, comments and corrections welcome.
