**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.

**If you find this page helpful, please Like or Share or Leave a Comment. Thank you! **

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)*. Plug in any (x, y) and solve for

^{x}*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) = -x ^{2} + 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 x^{2} 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 = -x^{2} + 8x

Factor out the -1 coefficient: y – 9 = -(x^{2} - 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 = -(x^{2} - 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 x^{2}.

The Area of the rectangular garden is A = (2x)(x – 3) = 2x^{2} - 6x. The Area of the rectangular garden is also 1.25x^{2}.

So 2x^{2} - 6x = 1.25x^{2}

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

Solving the equation: 2x^{2} - 6x = 1.25x^{2}

0.75x^{2} - 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. *

## 11 comments:

Great.....thank u very much....

But in the qus of 37...how us get the area of rectangular garden is 1.25x square??

It is 25% bigger, that makes it 125% of the size, which as a decimal is 1.25.

I don't get how you got 1.25. Shouldn't it be .25 since 25 divided by 100 is .25. Can you please explain how you got 1.25?

The area is increased by 25%. It was already 100%, so now it is 125% or 1.25.

ohh, thank you!

How do you get 0.75x^2

Subtract act 1.25x^2 from 2x^2 leaves 0.75x^2 on the left side and 0 on the right side.

In q 37 shouldn't the square have a side length of 8 not 4?

In q 37 shouldn't the square have a side length of 8 not 4?

Good catch. A typo, as it’s 8 in the equations. I’ll have to edit that in the morning.

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