June 2019 Algebra I, Part I
Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.
1. The expression w4 - 36 is equivalent to
Answer: (4) (w2 + 6)(w2 - 6)
Difference of Squares rule. You get the conjugates consisting of the square root of each term. ("Conjugates" means the same two terms, but one binomial has a plus and the other a minus.)
When multiplied the two w2 terms with negate each other. (Cancel out.)
2. If f(x) = 4x + 5, what is the value of f(-3)?
Answer: (2) -7
Substitute and evaluate
f(-3) = 4(-3) + 5 = -12 + 5 = -7
3. Which relation is not a function?
Answer: (4)
The map in (4) shows two arrows coming from 2 in the domain to two different elements of the range. This is not allowed in a function.
In Choice (1), no x value is repeated.
In Choice (2), the graph passes the vertical line test.
In Choice (3), you have a linear equation in standard form. All linear equations, with the exception of vertical lines (such as x = 3), are functions.
4. Given: f(x) = (x - 2)2 + 4
and g(x) = (x - 5)2 + 4
When compared to the graph of f(x), the graph of g(x) is
Answer: (2) shifted 3 units to the right
The vertex of f(x) is (2, 4).
The vertex of g(x) is (5, 4).
To get to g(x) from f(x), you have to shift 3 units to the right.
5. Students were asked to write 6x5 + Bx - 3x3 + 7x7 in standard form.
Shown below are four student responses.
Anne: 7x7 + 6x5 - 3x3 +Bx
Bob: -3x3 + 6x5 + 7x7 +Bx
Carrie: Bx + 7x7 + 6x5 - 3x3
Dylan: Bx - 3x3 + 6x5 + 7x7
Which student is correct?
Answer: (1) Anne
Standard form puts the term with the highest exponent first. Second highest exponent second, etc.
Anne is the only one to put the x7 term first.
6. The function f is shown in the table below
Which type of function best models the given data?
Answer: (1) exponential growth function
Quick elimination: there is not a constant rate of change, so it is not linear. So (3) and (4) are incorrect. The values are increasing, so it is not decay, and (2) is incorrect.
The f(x) values are increasing and have a common ratio, 3/1 = 9/3 = 27/9. It is an example of exponential growth.
More specifically, it is f(x) = 3x
7. Which expression results in a rational number?
Answer: (1) Sqrt(2) * Sqrt(18)
The sum of two irrational numbers will be irrational. The product of a rational number and an irrational number will be irrational.
The product of two irrational number could be rational, if, in this instance, it produces a perfect square under the radical sign. The square root of 36 is 6, which is rational.
8. A polynomial function is graphed below.
Which function could represent this graph?
Answer: (3) f(x) = (x - l)(x2 - 4)
The zeroes of the function are -2, 1, and 2.
The factors of the function must be (x + 2), (x - 1) and (x - 2).
The product of (x + 2)(x - 2) is (x2 - 4).
Looking at the choices, you know that there are three real roots.
Neither (x2 + 2) nore (x2 + 4) have real roots. They wouldn't touch the x-axis.
Choice (2) is no good because (x2 - 2) would have roots at the square root of 2, which would be in the middle of 1 and 2 (and -1 and -2) on the graph.
9. When solving p2 + 5 = Sp - 7, Kate wrote p2 + 12 = Sp. The
property she used is
Answer: (4) the addition property of equality
She added 7 to both sides of the equation. Associative property changes the groupings, which didn't happen. Commutative property changes the order, which didn't happen. The Distributive Property involves parentheses, which aren't there.
10. David wanted to go on an amusement park ride. A sign posted at the
entrance read "You must be greater than 42 inches tall and no more
than 57 inches tall for this ride." Which inequality would model the
height, x, required for this amusement park ride?
Answer: (1) 42 < x < 57
The variable has be in between 42 and 57. Only choice number (1) makes any sense.
Choice (2) literally says 42 > 57. Not true.
Choice (3) would work if it said AND instead of OR because, for example, 10 is less than 57, so that makes the entire condition true.
Choice (4) is almost, but not quite, the conditions for people cannot use the ride.
11. Which situation can be modeled by a linear function?
Answer: (3) An amusement park allows 50 people to enter every 30 minutes.
Every 30 minutes 50 people enter is a constant rate, so it is a linear function.
Choice (1) is exponential growth.
Choices (2) and (4) are exponential decay.
12. Jenna took a survey of her senior class to see whether they preferred
pizza or burgers. The results are summarized in the table below.
| | Pizza | Burgers |
| Male | 23 | 42 |
| Female | 31 | 26 |
Of the people who preferred burgers, approximately what percentage
were female?
Answer: (2) 38.2
26 / (26 + 42) = 0.38235...
Just looking at the numbers, you could tell that it's a little more than a third.
Only one choice is a little more than a third. Another is less than a third, and the next is almost one-half.
13. 2a - 8b is solved for a, the result is
Answer: (4) a > -l5b
3a + 7b > 2a - 8b
a + 7b > - 8b
a > -15b
14. Three functions are shown below:
A: g(x) = -3/2 x + 4
B: f(x) = (x + 2)(x + 6)
C: (graph)
Which statement is true?
Answer: (3) B has a minimum and C has a maximum.
B is a parabola that opens upward, so it has a minimum. The graph of C shows a maximum.
Choice (1) is incorrect because the zeroes of B are -2 and -6, but C shows -2 and -4.
Choice (2) is incorrect because the y-intercept of A is 4, but the y-intercept of B is (2)(6) = 12.
Choice (4) is incorrect because A has no minimum -- it is a linear function.
15. Nicci’s sister is 7 years less than twice Nicci’s age, a. The sum of
Nicci’s age and her sister’s age is 41. Which equation represents this
relationship?
Answer: (2) a + (2a - 7) = 41
7 less than twice a is 7 less than 2a, which is 2a - 7. It is NOT 7 - 2a.
(Consider: if I have 7 less than 100, I have 100 - 7 = 93, not 7 - 100 = -93.)
The sum of that expression and a is 41, so a + (2a - 7) = 41.
16. The population of a small town over four years is recorded in the chart
below, where 2013 is represented by x 0. [Population is rounded to
the nearest person]
| Year | 2013 | 2014 | 2015 | 2016 |
| Population | 3810 | 3943 | 4081 | 4224 |
The population, P(x), for these years can be modeled by the function
P(x) = abx, where b is rounded to the nearest thousandth. Which
statements about this function are true?
I. a = 3810
II. a = 4224
III. b = 0.035
IV. b = 1.035
Answer: (2) I and IV
Put the numbers in Lists and do an Exp regression. You'll get a = 3810 and b = 1.035.
WITHOUT PUTTING IT IN THE CALCULATOR: The first number is 3810 and the last is 4224. The base has to be 3810. This is an example of exponential growth, not decay, so the base has to be greater than 1.
17. When written in factored form, 4w2 - 11w + 3 is equivalent to
Answer: (3) (4w + 1)(w - 3)
Looking quickly, all four will give you 4w2 and -3. So you only have to concern yourself with the inner and outer terms.
(4w)(-3) = -12w and (1)(w) = w, -12w + w = -11w, which is what we want.
18. Which ordered pair does not represent a point on the graph of
y = 3x2 - x + 7?
Answer: (3) (1.25, 10.25)
If you plot this, you can manipulate the table of values to show you multiples of .25. Or you can use the Trace function to zero in on the points. You can also use a list of a set to check all four values at once.
Or just check one at a time:
3(-1.5)^2 - (-1.5) + 7 = 15.25
3(0.5)^2 - (0.5) + 7 = 7.25
3(1.25)^2 - (1.25) + 7 = 10.4375
3(2.5)^2 - (2.5) + 7 = 23.25
If you brute force it, don't stop at the first wrong answer. Keep going. If you get two wrong answers, you know you mis-entered something. Odds are low that you're accidentally get the "right" answer to appear by mistake.
19. Given the following three sequences:
I. 2, 4, 6, 8, 10...
II. 2, 4, 8, 16, 32...
III. a, a + 2, a + 4, a + 6, a + 8...
Which ones are arithmetic sequences?
Answer: (2) I and III, only
Each term in I increases by 2. Each term in II doubles the previous, which is geometric, not arithmetic. Each term in III is 2 more than the one before it.
20. A grocery store sells packages of beef. The function C(w)
represents the cost, in dollars, of a package of beef weighing w pounds. The most appropriate domain for this function would be
Answer: (4) positive rational numbers
Weight of beef does not have to be in whole number pounds, so integers is not appropriate.
Likewise, the weight have to be a non-negative number. While zero is a valid number for weight, it doesn't make sense for a sale -- you couldn't sell nothing. So positive rational numbers is the best.
21. The roots of x2 - 5x - 4 = 0 are
Answer: (2) (5 + SQRT(41) ) / 2
If either Choice (1) or (3) were correct, the final term would have to be + 4, not - 4.
(x - 1)(x - 4) and (x + 1)(x + 4) both have + 4 as the final term.
The quadratic formula (as the Axis of Symmetry formula) starts with -b.
A -(-5) is +5, which means the correct answer is (2) not (4).
22. The following table shows the heights, in inches, of the players on
the opening-night roster of the 2015-2016 New York Knicks.
| 84 | 80 | 87 | 75 | 77 | 79 | 80 | 74 | 76 | 80 | 80 | 82 | 82 |
The population standard deviation of these data is approximately
Answer: (1) 3.5
No calculations necessary. Okay, one small one.
The range of the data is 87 - 74 = 13.
The standard deviation is much smaller than that, and the only possible choice is (1).
Without doing any other work, I would estimate that 79.7 is the mean and 80, which is obviously the mode, is likely the median as well.
23. W A population of bacteria can be modeled by the function
f(t) = 1000(0.98)t, where t represents the time since the population started decaying, and f(t) represents the population of the remaining bacteria at time t. What is the rate of decay for this population?
Answer: (2) 2%
Two percent decay means that 98% of the bacteria remain, and that is written as 0.98.
24. Bamboo plants can grow 91 centimeters per day. What is the
approximate growth of the plant, in inches per hour?
Answer: (1) 1.49
You need to convert centimeters per day into inches per hour.
The reference sheet gives you 1 inch 2.54 centimeters, and, of course, there are 24 hours in one day.
So (91 cm / day) * (1 day / 24 hrs) * (1 in / 2.54 cm) = 1.49278...
End of Part I
How did you do?
Questions, comments and corrections welcome.
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