Monday, January 30, 2017

January 2017: Common Core Algebra Regents, Part 2

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.

January 2017, Algebra I (Common Core), Part II

25. In attempting to solve the system of equations y = 3x - 2 and 6x - 2y = 4, John graphed the two equations on his graphing calculator. Because he saw only one line, John wrote that the answer to the system is the empty set. Is he correct? Explain your answer.

John is not correct. The answer is not the empty set. The answer is that the two lines are coincident (they overlap each other), so there are an infinite number of points that solve this system of equations.

Proof: 6x - 2y = 4
-2y = -6x + 4
y = 3x - 2, which is the other line.
The two lines are coincidental lines.

The empty set would mean that the lines never intersected; i.e., they are parallel and have no points in common.

26. A typical marathon is 26.2 miles. Allan averages 12 kilometers per hour when running in marathons. Determine how long it would take Allan to complete a marathon, to the nearest tenth of an hour. Justify your answer.

"Justify" means that you need to show your work or offer some kind of proof.
This problem is about unit conversion, and the conversion you need is in the back of the book: 1 mile = 1.60943 km
26.2 miles = 26.2(1.60943 km/mi) = 42.164708 km
42.164708 km / (12 km/hr) = 3.5137256666....
Which is 3.5 to the nearest tenth of an hour.

27. Solve the inequality below: 1.8 - 0.4y > 2.2 - 2y

Solve inequalities like you solve equations, but be mindful of dividing by negative numbers.

1.8 - 0.4y > 2.2 - 2y
-0.4y > 0.4 - 2y
-0.4y > 0.4 - 2y
1.6y > 0.4
y > 0.25

Unusual that they didn't multiply or divide by a negative, but watch out for it.

28. Jakob is working on his math homework. He decides that the sum of the expression 1/3 + 6*sqrt(5)/7 must be rational because it is a fraction. Is Jakob correct? Explain your reasoning.

Jakob is not correct. For a fraction to be rational, both the numerator and the denominator must be rational numbers. Six radical Five is an irrational number, and dividing it by 7 doesn't make it rational.

Finally, the sum of any rational number (e.g., 1/3) and an irrational number is always irrational.

29. Graph the inequality y > 2x - 5 on the set of axes below. State the coordinates of a point in its solution.

The graph will be a broken line with a slope of 2 and a y-intercept of -5 that is shaded ABOVE the line.
If you don't remember where to shade, here is the simple test: take the point (0, 0) and substitute it into the inequality. If the result a true statement? Is 0 > 2(0) - 5? Yes, 0 > -5, so that point is in the shaded region. Because that point is above the line, you will shade above the line. Like this:

Remember to state ANY point in the solution. (0, 0) would work. Or (-10, 0). Or (0, 10). Any point in the shaded section that is NOT on the line. The line is NOT part of the solution because it is broken.

30. Sandy programmed a website's checkout process with an equation to calculate the amount customers will be charged when they download songs. The website offers a discount. If one song is bought at the full price of $1.29, then each additional song is $.99. State an equation that represents the cost, C, when s songs are downloaded.
Sandy figured she would be charged $52.77 for 52 songs. Is this the correct amount? Justify your answer.

The linear equation that models this problem is (Note that the variables are up to you): C = 1.29 + .99(n - 1)
You pay $1.29 for the first song. If you buy n song, you will pay 99 cents for n - 1 of them.

To find the cost of 52 songs, substitute 52 for n: C = 1.29 + .99(52 - 1) = 51.78
Sandy is incorrect.

Not that they asked this, but it looks like her mistake is that she calculated C = 1.29 + .99(52), without subtracting 1 first.

31. A family is traveling from their home to a vacation resort hotel. The table below shows their distance from home as a function of time.

Time (hrs) 0 2 5 7
Distance (mi) 0 140 375 480

Determine the average rate of change between hour 2 and hour 7, including units.

Average rate of change between points (2, 140) and (7, 480) is the difference of the y values divided by the difference of the x values:
(480 - 140) / (7 - 2) = 340 / 5 = 68 mi / hr.

Note that the question said to include units, so you better include the units or you will lose a point!

32. Nora says that the graph of a circle is a function because she can trace the whole graph without picking up her pencil.
Mia says that a circle graph is not a function because multiple values of x map to the same y-value.
Determine if either one is correct, and justify your answer completely.

Neither one is correct. Nora is wrong because not lifting a pencil is not the definition of a function. Piecewise functions sometimes require you to lift your pencil. Mia is wrong because she has it backwards: there are multiple values of y for the same x, which is the vertical line test. What Mia described would be a horizontal line test, but there is no such thing for functions. You are allowed to have multiple values of x map to the same value of y. (For example: the absolute value function repeats y values for more than one x.)

End of Part II

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.


Anonymous said...

It helped a lot, thanks. I had trouble with the questions so I came to this site to check my work.

(x, why?) said...

You're welcome. That's why I do this.

Unknown said...
This comment has been removed by the author.
Anonymous said...

Thank you. I was so stuck and this was helpful.