(x, why?): Algebra Problems of the Day (Algebra Regents, August 2024 Part I)

This exam was adminstered in August 2024.

August 2024 Algebra Regents

Part I

Each correct answer will receive 2 credits. No partial credit.

9. The heights, in inches, of eight football players are given below.

76, 70, 72, 70, 69, 71, 78, 74 Which box plot represents these data?

Answer: (3) [see image]

First, put the numbers in order: 69, 70, 70, 71, 72, 74, 76, 78.

The 5-Number Summary is Min 69 and Max 78.
You can eliminate Choices (1) and (2).

The median is (71+72)/5 = 71.5.
Eliminate Choice (4).

Q1 is 70 and Q3 is (74+76)/2 = 75.

The correct answer is Choice (3).

10. A bookstore owner recorded the number of books sold and the profit made selling the books.

What is the average rate of change, in dollars per book, between 100 and 350 books sold?

(1) 0.50
(2) 0.67
(3) 1.50
(4) 2.00

Answer: (3) 1.50

Subtract the profit for 100 books from the profit for 350 books and divide the difference by 250 (which is 350 - 100).

(425 - 50) / 250 = 1.5, which is Choice (3).

11. If f(x) = x², then which function represents a shift of the graph of f(x) 4 units to the right and 3 units down?

(1) g(x) = (x + 4)² + 3
(2) j(x) = (x + 4)² - 3
(3) h(x) = (x - 4)² - 3
(4) k(x) = (x - 4)² + 3

Answer: (3) h(x) = (x - 4)² - 3

Subtracting inside the parentheses moves the parabola to the right. (Adding moves it to the left.)

Subtracting outside the parentheses moves the parabola down. (Adding movies it up.)

Choice (3) is the correct answer.

12. The amount of money a plumber charges is represented by the function p(h) = 45 + 90h. The best interpretation of the y-intercept of this function is that the plumber charges

(1) $45 to come to the house
(2) $45 per hour that he works
(3) $90 to come to the house
(4) $90 per hour that he works

Answer: (1) $45 to come to the house

The plumber charges $45 to come to the house and $90 per hour that he works, so $45 would be the y-intercept and $90 would be the slope of the graph of this function.

Choice (1) is the correct answer.

13. What is the solution to the inequality 2m - 4 < 3(2m + 4)?

(1) m < -2
(2) m > -2
(3) m < -4
(4) m > -4

Answer: (4) m > -4

2m - 4 < 3(2m + 4)
2m - 4 < 6m + 12
-4 < 4m + 12
-16 < 4m
-4 < m
m > -4

Choice (4) is the correct answer.

14. A survey of students at West High School was taken to determine a theme for the prom. The results of the survey are summarized in the table below.

Approximately what percentage of the students who chose the Broadway theme were girls?

(1) 26
(2) 27
(3) 46
(4) 68

Answer: (3) 46

Divide 68 by the number who chose Broadway, which is (68 + 79) = 147.

68/147 = 0.462..., or about 46%.

Choice (3) is the correct answer.

15. The sum of 2√(54) and 2√(6) is

(1) 4√(60)
(2) 8√(15)
(3) 7√(6)
(4) 8√(6)

Answer: (4) 8√(6)

Simplify the radicals so that they can be added together. You need to have a common radical, similar to needing a common denominator for fractions. Basically, you want to combine like terms.

2√(54) + 2√(6)
2√((9)(6)) + 2√(6)
3*2√(6) + 2√(6)
= 8√(6)

Choice (4) is the correct answer.

16. The functions f(x) = x² - 5x - 14 and g(x) = x + 2 are graphed on the same set of axes. What are the solutions to the equation f(x) 5 g(x)?

(1) -14 and 0
(2) 0 and 2
(3) -2 and 8
(4) -2 and 7

Answer: (3) -2 and 8

You can put these functions into the graphing calculator and check the Table of Values to find where the values are equal. Otherwise, you have to solve for x:

x² - 5x - 14 = x + 2
x² - 6x - 16 = 0
(x - 8)(x + 2) = 0
x - 8 = 0 or x + 2 = 0
x = 8 or x = -2

The correct answer is Choice (3).

More to come. Comments and questions welcome.

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