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

This exam was adminstered in August 2024 .

August 2024 Algebra, Part III

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

31. The owner of an ice cream stand kept track of the number of ice cream cones that were sold each day of the first week in June. She compared the ice cream sales to the average daily temperature. The data are shown in the table below.

State the linear regression equation for these data, rounding all values to the nearest hundredth.

State the correlation coefficient, to the nearest hundredth, for the line of best fit for these data.

State what this correlation coefficient indicates about the linear fit of the data.

Answer:

You have to perform a Linear Regression in your calculation. Put the data from the table into two lists (L1 and L2).

After rounding the values in your calculator, you will get the equation y = 15.13x - 959.63.

The correlation coefficient, r, is 0.99.

This correlation coefficient indicates that there is a strong postivie correlation in the data.

32. Graph the system of inequalities on the set of axes below:

y > 3x - 4
x + 2y <6
Label the solution set S.
Is the point (2,2) a solution to the system? Justify your answer.

Answer:

Remember that > means a broken line with shading about the line and that < means a solid line with shading below that line.

You can use your calculator to graph the lines and use the Table of Values. Remember that the first line must be broken or dotted.

Rewrite the second equation as y < -1/2 x + 3.

Your graph will look like this:

The point (2,2) is NOT is the solution set of the system of inequalities because it is on the broken line. The points on the broken line are NOT part of the solution.

Note: your final answer will depend upon the graph you draw. If you have a graphing error, then the answer to the final part of the question MUST match your graph.

33. An object is launched upward at 64 feet per second from a platform 80 feet above the ground. The function s(t) models the height of the object t seconds after launch.

If s(t) = -16t² + 64t + 80, state the vertex of s(t), and explain in detail what each coordinate means in the context of the problem.

After the object is launched, how many seconds does it take for the object to hit the ground? Justify your answer.

Answer:

This is a problem of gravity, which creates an upside-down parabola when graphed. The vertex is a point on the Axis of Symmetry, which can be found using x = -b/(2a).

The Axis of Symmetry is x = -64/(2(-16)) = 2.

Calculate s(2) = -16(2)² + 64(2) + 80 = 144. The coordinates of the vertex are (2,144).

In the context of this problem, that point says that at two seconds the object reaches its maximum height, which is 144 feet above the ground.

The object will hit the ground at when -16t² + 64t + 80 = 0.

-16t² + 64t + 80 = 0
t² - 4t - 5 = 0
(t - 5)(t + 1) = 0
t - 5 = 0 or t + 1 = 0
t = 5 or t = -1

Discard the negative answer because time cannot be negative. The object hits the ground at t = 5 seconds.

34. Solve the system of equations algebraically for all values of x and y.

y = x² + 4x - 1
y = 2x + 7

Answer:

Set the two expressions equal to each other. Solve the quadratic equation. Then solve for y.

x² + 4x - 1 = 2x + 7
x² + 2x - 8 = 0
(x + 4)(x - 2) = 0
x + 4 = 0 or x - 2 = 0
x = -4 or x = 2

If x = -4, y = (-4)² + 4(-4) - 1 = -1.

If x = 2, y = (2)² + 4(2) - 1 = 11.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

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

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.