Algebra Problems of the Day (Algebra Regents, August 2024 Part II)

This exam was adminstered in August 2024 .

August 2024 Algebra, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. On the set of axes below, graph f(x) = x² + 4x + 1.
State the coordinates of the minimum.

Answer:

You can use your graphing calculator and check the Table of Values. Be sure to state the coordinates of the minimum. Label it if you have any other points listed.

The minimum point is (-2, -3).

26. If f(x) = 30x²/(x + 2), determine f(1/2).

Answer:

Substitute and evaluate. Remember the order of operations. Or put it into your calculator. (If anyone is reading all of my Algebra posts, you might have noticed that I wrote the exact same thing for Question 26 on the June 2024 Regents becuase they asked a Very Similar Question!)

f(1/2) = 30 (1/2)² / (1/2 + 2) = (30)(1/4) / 2.5 = 3

If you put this into your calculator, remember to use all the parentheses that I used above to avoid a mistake.

You could also graph y = 30x²/(x + 2) and check the table of values for x = .5.

Whatever you do, write something on the paper other than just the answer, which is only worth one point. I suggest you write down the substition. It will be assumed that you calculated it in your head or with a calculator, so you don't need to write down every step.

27. Explain why the relation shown in the table below is a function.

Complete the table below with values for both x and y so that this new relation is not a function.

Answer:

The relation is a function as long as the x values do not repeat. (Again! I wrote this very same comment in June for Question 27!) Since the x values don't repeat, it is a function.

In the second table, repeat any of the values of the first line in the first box. In the second box, write a different number. In other words, 0 and 6 is a good answer, but 0 and 4 is not because (0,4) is already in the relation. 2 and 4 is a good answer, but 2 and 5 is not.

28. Solve algebraically for x: 0.05(x - 3) = 0.35x - 7.5

Answer:

Multiply the equation by 20 to get rid of the 0.05. (Or divide everything by 0.05, which is the same thing.) Then move the variables to one sides and the numbers to the other.

0.05(x - 3) = 0.35x - 7.5
x - 3 = 7x - 150
147 = 6x
24.5 = x

29. Use the quadratic formula to determine the exact roots of the equation x² + 3x - 6 = 0.

Answer:
If it says to use the quadratic formula, then you must use it. If you use any other method, then the most you can get is 1 point for a correct value of x.

The quadratic formula is x = (-b + √(b² - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = 3, and c = -6.

x = (-b + √(b² - 4ac)) / (2a)
x = (-3 + √(3² - 4(-6))) / (2)
x = (-3 + √(9 + 24)) / (2)
x = -3/2 + √(33) / 2

Those are the exact values. Do not approximate by changing them to decimals.

30. Factor 5x³ - 80x completely.

Answer:
Whenever you see "Factor completely", you can be pretty sure that there will be more than one step. Show them all. Once again, I copied text from Question 30 of the June Regents!)

First, look for common factors in each of the terms. Then factor what remains. There are two terms separated by a minus sign, so keep an eye out for a Difference of Squares.

5x³ - 80x = 5x (x² - 16) = 5x (x + 4)(x - 4)

Note that this is an expression and not an equation. Do NOT attempt to "solve" it, or you will lose a point.

End of Part II

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