## Friday, July 26, 2024

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

This exam was adminstered in June 2024.

More Regents problems.

### Part I

9. The zeros of the function f(x) = x(x - 5)(3x + 6)

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

The zeroes are the values x that make the function equal to zero.

Since the first of the three terms that are being multiplied is just x, then the funtion is equal to 0 when x = 0, because zero times anything is zero. Eliminate Choices (3) and (4).

The second term (x - 5) is equal to 0 when x = 5 because (5 - 5) = 0. Eliminate Choice (1).

Choice (2) is the answer because (5 - 5) = 0 and (3(-2) + 6) = 0.

10. What is the y-intercept of the line that passes through the points (-1,5) and (2,-1)?

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

The y-intercept is the value of y when x = 0, or the y-coordinate when the line crosses the y-axis.

Before you do anything, it should be obvious that the number must be in-between 5 and -1, so eliminate Choices (1), (2), and (4).

Choice (3) is the only possible Choice and is the correct answer.

If you didn't realize this, you can use the scratch graph paper to plot the points and draw the line, using a straightedge or ruler, to find the answer.

You could also find the slope of the line: (-1 - 5) / (2 - -1) = -6 / 3 = -2.

If the slope is -2, then the point after (-1, 5) is (0, 3), which is Choice (3).

11. Nancy has just been hired for her first job. Her company gives her four choices for how she can collect her annual salary over the first eight years of employment.
Each function below represents the four choices she has for her annual salary in thousands of dollars, where t represents the number of years after she is hired.

a(t) = 2t + 25
b(t) = 10t + 75
c(t) = √(400t) + 80
d(t) = 2(t + 1)2 - 10t + 50

Which pay plan should Nancy choose in order to have the highest salary in her eighth year?

(1) a(t)
(2) b(t)
(3) c(t)
(4) d(t)

First, an important note to all high school students taking this exam: THIS WILL NEVER HAPPEN IN REAL LIFE!

They are asking which will be the highest amount a(8), b(8), c(8) or d(8). That's it. Forget about salaries and jobs. Just put them in the calculator, and be sure that you entered them correctly when it comes to the exponents and square roots.

a(8) = 28 + 25 = 281

b(8) = 10(8) + 75 = 155

c(8) = √(400(8)) + 80 = 136.57

d(8) = 2(8 + 1)2 - 10(8) + 50 = 132

The maximum amount is Choice (1), a(t), which you might have suspected because it is the only one that is exponential, and exponential functions rise more quickly than other functions. But it doesn't hurt to check.

12. The third term in a sequence is 25 and the fifth term is 625. Which number could be the common ratio of the sequence?

(1) 1/5
(2) 5
(3) 1/25
(4) 25

The sequence is increasing, so the common ratio must be greater than 1. Eliminate Choices (1) and (3).

If you multiply, 25 * 5 = 125 and 125 * 5 = 625, you have the correct answer. Choice (2) is the correct choice.

If you multiply 25 * 25 = 625, then the third term is 625, not the fourth term.

How you're supposed to solve this, if you didn't have choices to work with, is as follows:

25 * x is term 3, and term 3 * x = 625, so 25 x2 = 625.
That means that x2 = 25, and x = 5.

13. The box plot below summarizes the data for the amount of snowfall, in inches, during the winter of 2021 for 12 locations in western New York.

What is the interquartile range?

(1) 30
(2) 50
(3) 80
(4) 110

The interquartile range is the difference between the third or upper quartile (Q3) and first or lower quartile (Q1). Basically, it's the length of the box.

Q3 = 110, and Q1 = 60, and 110 - 60 = 50, so the IQR = 50, which is Choice (2).

Choice (1), 30, is the length from the median to Q3. Eliminated Choice (1).

Choice (3), 80, is the median of the data, not the length of the box. Eliminate Choice (3).

Choice (4), 110, is Q3, not the interquartile range. Eliminate Choice (4).

14. Four quadratic functions are represented below.

Which function has the smallest minimum value?

(1) I
(2) II
(3) III
(4) IV

You can check the minimum value of function c(x) in your calculator, or you can work it out.

Function a(x) is in vertex form, meaning that (3,-7) is the vertex, so the minimum value for a(x) = -7. The minimum value of b(x) is -4 and the minumum value of d(x) is -5. Eliminate Choices (2) and (4).

The minimum value of c(c) is on the Axis of Symmetry, which is the line with the equation x = -b/(2a) = -(6)/(2) = -3. The value of c(-3) = (-3)2 + 6(-3) + 3 = -6. Eliminate Choice (3).

Choice (1), I, is the correct answer.

15. The equation that represents the sequence -2, -5, -8, -11, -14, . . . is

(1) an = -3 + (-2)(n - 1)
(2) an = -2 + (-3)(n - 1)
(3) an = 3 + (-2)(n - 1)
(4) an = -2 + (3)(n - 1)

Answer: (2) an = -2 + (-3)(n - 1)

The initial term is -2 and the common difference is -3. The Choice that shows this is Choice (2), which is the Correct answer.

Check: Let's let n = 4 to find the value of the fourth term

-3 + (-2)(4 -1) = -9, not -11. Eliminate Choice (1).
-2 + (-3)(4 - 1) = -11. This is correct.
3 + (-2)(4 - 1) = -3, not -11. Eliminate Choice (3).
-2 + (3)(4 - 1) = 7, not -11. Eliminate Choice (4).

Only Choice (2) gives the correct fourth term.

16. The dot plot below shows the number of goals Jessica scored in each lacrosse game last season.

Which statement about the dot plot is correct?

(1) mean > mode
(2) mean = median
(3) mode = mediam
(4) median %gt; mean

The mode is 3, because there are 5 of them. The median is 2.5 because there are 20 pieces of data and the 10th is 2 and the 11th is 3, so the median is 2.5. The median does not equal the mode, so eliminate Choice (3).

The other three choices use the mean, so you have to calculat the mean, which is the average of all the data. That is, add up all 20 numbers and divide by 20. You can take a shortcut by using multiplication for each individual score. Obviously, I don't need to add the zeros, but I'll write it down just for completeness.

(3 * 0) + (3 * 1) + (4 * 2) + (5 * 3) + (2 * 4) + (2 * 5) + (1 * 6) = 50
50 / 20 = 2.5

The mean is not greater than the mode, so Choice (1) is eliminated.

The mean is 2.5 which is equal to the median. The correct answer is Choice (2).

More to come. Comments and questions welcome.

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