## Monday, July 29, 2024

### Algebra Problems of the Day (Algebra Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

### Part I

17. The students in Mrs. Smith’s algebra class were asked to describe the graph of g(x) = 2(x - 3)2 compared to the graph of f(x) = x2.
Which student response is correct?

(1) Ashley said that the graph of g(x) is wider and shifted left 3 units.
(2) Beth said that the graph of g(x) is narrower and shifted left 3 units.
(3) Carl said that the graph of g(x) is wider and shifted right 3 units.
(4) Don said that the graph of g(x) is narrower and shifted right 3 units.

Answer: (4) Don said that the graph of g(x) is narrower and shifted right 3 units.

If the leading coefficient of a quadratic function is greather than 1, then the parabola will become skinnier, or narrower. (I like to say "skinnier" because I have trouble saying "narrower" in a classroom setting. I don't know why -- I just do.) Eliminate Choices (1) and (3) which happen when the coefficient is between 0 and 1.

If you subtract three from x inside the parentheses, the entire graph will shift to the right. So Choice (4) is the correct response.

18. One Saturday, Dave took a long bike ride. The graph below models his trip.

What was Dave’s average rate of change, in miles per hour, on this trip?

(1) 10
(2) 11
(3) 11.6
(4) 14.5

The average rate of change in miles per hour is literally the number of miles he traveled divided by the number of hours he traveled.

The final coordinate is (5.5, 55). Divide 55/5.5 = 10, which is Choice (1).

11. Which expression is equivalent to (x - 5)(2x + 7) - (x + 5)?

(1) 2x2 - 2x - 30
(2) 2x2 - 2x - 40
(3) 2x2 - 4x - 30
(4) 2x2 - 4x - 40

Answer: (4) 2x2 - 4x - 40

If you multiply (-5)(7), you get -35, then subtract +5, you get -40. That means that Choices (1) and (3) can be eliminated.

(x - 5)(2x + 7) - (x + 5)
2x2 + 7x - 10x - 35 - x - 5
2x2 - 4x - 40

Choice (4) is the correct answer.

20. The functions f(x) and g(x) are graphed on the set of axes below.

What is the solution to the equation f(x) = g(x)?

(1) 1 and 5
(2) -5 and 0
(3) -3 and 5
(4) 0 and 4

For which values of x will f(x) = g(x). They do NOT want the values of f(x) or g(x). They want x.

Look for the points on the graph where the two functions intersect. Do not look at the points where the lines cross the x-axis -- that's a different kind of problem.

Looking at the graph, the lines intersect at x = 1 and x = 5. This is Choice (1).

Choice (2) is the y-intercepts of the two graphs.

Choice (3) is the values of f(x) and g(x), which isn't the question.

Choice (4) is the roots of f(x) and has nothing to do with g(x).

21. When babysitting, Nicole charges an hourly rate and an additional charge for gas. She uses the function C(h) = 6h + 5 to determine how much to charge for babysitting. The constant term of this function represents

(1) the additional charge for gas
(2) the hourly rate Nicole charges
(3) the number of hours Nicole babysits
(4) the total Nicole earns from babysitting

The fee for gas is the constant, which is Choice (1).

The hourly rate is 6, which is the coefficient of the variable (the slope of the line, the rate of change).

The number of hours Nicole babysits is the variable.

The total Nicole earns is the value of C(h).

22. When solved for x in terms of a, the solution to the equation 3x - 7 = ax + 5 is

(1) 12 / (3a)
(2) 12 / (3-a)
(3) 3a / 12
(4) (3 - a) / 12

Get all of the x terms on one side of the equation, and then isolate the variable.

3x - 7 = ax + 5
3x - ax = 5 + 7
x(3 - a) = 12
x = 12 / (3 - a)

This is Choice (2).

23. Wayde van Niekerk, a runner from South Africa, ran 400 meters in 43.03 seconds to set a world record. Which calculation would determine his average speed, in miles per hour?

Answer: (3) 400 m / 43.04 sec * 0.62 mi / 1000 m * 1 hr / 3600 sec

Units can be multipled and canceled like numerical factors can. When all the factors are canceled, only miles (on top) and hours (on the bottom) should remain.

In Choice (1), neither m nor sec cancel. It says m2hr / (mi * sec2). Eliminate Choice (1).

In Choice (2), sec doesn't cancel. It say mi * hr / (m * sec2). Eliminate Choice (2).

In Choice (3), both m and sec cancel, leaving mi/hr. This is the correct answer.

In Choice (4), m doesn't cancel. It say m2 * sev / (sec * mi * hr). Eliminate Choice (4).

24. Which function has a domain of all real numbers and a range greater than or equal to three?

(1) f(x) = -x + 3
(2) g(x) = x2 + 3
(3) h(x) = 3x
(4) m(x) = |x + 3|

The domain of all these function is all real numbers. None of these functions are undefined for any value of x.

To have a range of greater than or equal to 3, no output of the funtion can be less than 3.

f(1) = -1 + 3 = 2. Eliminate Choice (1).

h(0) = 30 = 1. Eliminate Choice (3).

m(-3) = |-3 + 3| = 0. Eliminate Choice (4).

g(x) = (x - 0)2 + 3 have a vertex, the minimum point, at (0,3). The range must be 3 or greater. This is the correct answer.

Another way to look at it: Choice (1) is a linear equation that goes down to negative infinity. It has a range of all real numbers. Choice (2) is an exponential function that has an asymptote at y = 0 (the x-axis), so the range is greater than 0. Choice (4) is an absolute value function, so its range is greater than or equal to 0.

More to come. Comments and questions welcome.

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