January 2015 Common Core Algebra Regents, Parts III & IV

Update: I now have a Common Core Regents Review books available on Amazon.

If you missed the previous post, here is Part I, and here is most of Part II.

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The rest of the Common Core exam follows. It contains the last question from Part II and all of Parts III and IV.

32. Write an exponential equation for the graph shown below.

Explain how you determined the equation.

Exponential equations are in the form y = a(b)^x. There are two methods to get the correct equation.
First: Label the points that we know, not the ones you have to guess at.
These are (2, 1), (3, 2), (4, 4), (5, 8). We can see that as x increases by 1, the value of y doubles. So the base (b) is 2. This gives us y = a(2)^x. Plug in any (x, y) and solve for a, which is ¼ or .25. Either is acceptable. The answer is y = .25(2)^x.

Second: put those same points into two LISTs on your graphing calculator. L1 = {2, 3, 4, 5}. L2 = {1, 2, 4, 8}. In the CALC menu, select ExpReg, for Exponential Regression. It will give you the same answer. EXPLAIN on your paper that you used the calculator and what steps you took (as I just did), and you will receive full credit. If you give the equation and just say “The calculator said so” then you will NOT get full credit.

Part III – Four points each

33. Jacob and Zachary go to the movie theater and purchase refreshments for their friends. Jacob spends a total of $18.25 on two bags of popcorn and three drinks. Zachary spends a total of $27.50 for four bags of popcorn and two drinks.

Write a system of equations that can be used to find the price of one bag of popcorn and the price of one drink.

Using these equations, determine and state the price of a bag of popcorn and the price of a drink, to the nearest cent.

Define your variables: p = cost of 1 popcorn, d = cost of 1 drink
Write an equation for each purchase:
2p + 3d = 18.25
4p + 2d = 27.50.

If we double the first equation, we can use the Elimination Method to solve this System of Equations.
4p + 6d = 36.50
4p + 2d = 27.50 (subtract the second equation from the first)
4d = 9.00
d = 2.25. The price of a drink is $2.25. Substitute in an original equation.
4p + 2(2.25) = 27.50
4p + 4.50 = 27.50
4p = 23.00
p = 5.75. The price of a popcorn is $5.75.

34. The graph of an inequality is shown below.

a) Write the inequality represented by the graph.

b) On the same set of axes, graph the inequality x + 2y < 4.

c) The two inequalities graphed on the set of axes form a system. Oscar thinks that the point (2,1) is in the solution set for this system of inequalities. Determine and state whether you agree with Oscar. Explain your reasoning.

a. The y-intercept of the graph is -3. The slope is +2 (2 up, 1 over). The line is solid, not broken and it is shaded above the line, which means > (greater than or equal to).
y > 2x – 3.

b. If you don’t know how to graph an inequality is Standard Form, you can rewrite it in Slope-Intercept Form, which is y < – ½ x + 2. The graph will look like this:

c. Oscar is incorrect because the (2, 1) is on a broken line. Points on the broken line are NOT part of the solution. They are a boundary to the solution. Everything up to, but NOT including, the line is a solution. NOTE: Your answer to this question depends upon the work you did before it. If you made an earlier mistake, then part c has to be consistent with that mistake.

35. A nutritionist collected information about different brands of beef hot dogs. She made a table showing the number of Calories and the amount of sodium in each hot dog.

a) Write the correlation coefficient for the line of best fit. Round your answer to the nearest hundredth.

b) Explain what the correlation coefficient suggests in the context of this problem.

The only way to do this is with your graphing calculator. Put the first column in L1. Put the second column in L2. In the CALC menu, select LinearReg. You will get the equation of the trend line along with the correlation coefficient, which is always a number between -1 and 1.

The correlation coefficient, rounded to two decimal places, is 0.94.

This means that there is a strong positive correlation between the two sets of data.

36. a) Given the function f(x) = -x² + 8x + 9, state whether the vertex represents a maximum or minimum point for the function. Explain your answer.

b) Rewrite f(x) in vertex form by completing the square.

Because the coefficient of x² is negative, the parabola opens downward, which means that the graph has a top or a maximum point. (It goes down forever, so there is no bottom, no minimum.)

Complete the square as follows: Subtract 9 from each side: y – 9 = -x² + 8x
Factor out the -1 coefficient: y – 9 = -(x² - 8x)
Complete the square by halving (-8) and squaring it. Add +16 inside the parentheses on the right side. On the left side add (-1)(16), or -16.
y – 9 – 16 = -(x² - 8x + 16)
Simplify the left side. Rewrite the right side as a binomial squared:
y – 25 = -(x – 4)²
Add 25 to both sides: y = -(x – 4)² + 25.
(FYI: The vertex is (4, 25) – but that wasn’t asked.)

Part IV – Six points

37. New Clarendon Park is undergoing renovations to its gardens. One garden that was originally a square is being adjusted so that one side is doubled in length, while the other side is decreased by three meters.

The new rectangular garden will have an area that is 25% more than the original square garden. Write an equation that could be used to determine the length of a side of the original square garden.

Explain how your equation models the situation.

Determine the area, in square meters, of the new rectangular garden.

Let x = the length of the square garden. Then the Area of the square garden is x².
The Area of the rectangular garden is A = (2x)(x – 3) = 2x² - 6x. The Area of the rectangular garden is also 1.25x².

So 2x² - 6x = 1.25x²

This equation models the situation because the new area is 25% bigger that the original, which is 1.25x², and the product of the new length times the new width is 2x² - 6x.

Solving the equation: 2x² - 6x = 1.25x²
0.75x² - 6x = 0
x(0.75x – 6) = 0
x = 0 or .75x – 6 = 0
.75x = 6
x = 8
We discard the 0 result because a square can’t have a length of 0, so the square had a side length of 8. The Area of the rectangular garden is 1.25(8)² = 1.25(64) = 80 square meters.

Edit: While the answer and the work all showed that x = 8, there was a typo in the sentence "so the square had a side length of", which has been fixed. I apologize for the inconvenience.