Wednesday, August 11, 2021

Algebra Problems of the Day (Integrated Algebra Regents, June 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part III: Each correct answer will receive 3 credits. Partial credit is possible.


34. On the set of axes below, graph the equation y = x2 + 2x - 8.

Using the graph, determine and state the roots of the equation x2 + 2x - 8 = 0.

Answer:


If you don't make the graph, the best you can get is 1 credit out of 3 for the roots.

You graphing calculator can give you a Table of Values, or you can compute them yourself. You know that the Axis of Symmetry is x = -b/(2a) = -2/2 = -1, and all the points away from that one will be mirror images.

Points on the graph looks like this:
y = (-1)2 + 2(-1) - 8 = -9, (-1, -9)
y = (0)2 + 2(0) - 8 = -8, (0, -8) and (-2, -8)
y = (1)2 + 2(1) - 8 = -5, (1, -5) and (-3, -5)
y = (2)2 + 2(2) - 8 = 0, (2, 0) and (-4, 0)
y = (3)2 + 2(3) - 8 = 7, (3, 7) and (-5, 7).

You may have noticed that the points go up by 1, then 3, then 5, then 7, ...

The graph looks like this:

The roots of the equation are -4 and 2.

Yes, you must state it, you can't just leave it on the graph. Do NOT say that the answer is (-4, 0) and (2, 0). Those are points on the graph and they are looking for solutions to a one-variable equation. Seriously.





35. A 28-foot ladder is leaning against a house. The bottom of the ladder is 6 feet from the base of the house. Find the measure of the angle formed by the ladder and the ground, to the nearest degree.

Answer:


The ladder leaning against the house forms a right triangle when you include the ground. The length of the ladder is the hypotenuse, and the distance from the house, 6 feet, is the side adjacent to the angle we are interesting in. That means that we have to use cosine.

Cos x = adj / hyp = 6/28

x = cos-1 (6/28) = 77.6 = 78 degrees

Don't forget to round to the nearest degree. And don't forget to put the calculator in Degree mode if it's in Radians mode.





36. Express ( 3 SQRT(75) + SQRT(27) ) / 3 in simplest form.


Answer:


If you are copying my answers for your homework, please make sure you use the radical symbols and not the letters SQRT, which are a holdover from computer programming. It'll be less frustrating for your teachers. Thank you.

Simplify the radicals by finding the largest perfect sqaures that are factors. Or just factor the and eliminate the doubles by taking the square root.

( 3 SQRT(75) + SQRT(27) ) / 3

= ( 3 SQRT(3 * 5 * 5) + SQRT(3 * 3 * 3) ) / 3

= ( 3 * 5 * SQRT(3) + 3 * SQRT(3) ) / 3

= ( 15 * SQRT(3) + 3 * SQRT(3) ) / 3

= ( 18 * SQRT(3) ) / 3

= 6 * SQRT(3)




End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

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