August 2015 Algebra 1 (Common Core) Regents: Part 2, Open-Ended

Here are the questions, with answers and explanations, for the New York State Algebra 1 (Common Core) Regents exam, Part 2. There were 8 questions, each worth 2 credits. Partial credit may be earned for correct work on a problem without a solution, or for a problem with a solution that contains one computational or conceptual error. All work must be shown. In general, a correct answer without any work is worth 1 credit, unless that answer is given as a choice and an explanation is required.

Link to Part 1

Part 2

25. Each day Toni records the height of a plant for her science lab. Her data are shown in the table below

The plant continues to grow at a constant daily rate. Write an equation to represent h(n), the height of the plant on the nth day.

The plant is growing at a constant rate, so use any two points to find the slope. Let's use (1, 3.0) and (2, 4.5). m = (4.5 - 3)/(2 - 1) = 1.5/1 = 1.5
Solve for b: 3 = 1.5(1) + b; b = 3 - 1.5 = 1.5
The function is h(n) = 1.5n + 1.5

Even if the slope or the y-intercept were obvious to you, show the work anyway. It helps.

26. On the set of axes below, graph the inequality 2x + y > 1.

Things they will be looking for: the correct slope and y-intercept, a broken line, proper shading.
In slope-intercept form: 2x + y > 1 becomes y > -2x + 1. Shade above the line.

27. Rachel and Marc were given the information shown below about the bacteria growing in a Petri dish in their biology class.

Rachel wants to model this information with a linear function. Marc wants to use an exponential function. Which model is the better choice? Explain why you chose this model.

The better choice is exponential because the function isn't growing at a constant rate.
If you find the rate of change for each pair of numbers, you will get the following: 60, 70, 80, 90, 110, 140, 170, 210, 270, 340. Not only is it NOT constant, it's growing with each passing hour.

If you divide B(2)/B(1), you get 1.27... Divide B(3)/B(2), you get 1.25. B(4)/B(3), you get 1.257.
You probably don't need to go all the way to 10 -- it's only a 2-point question -- but you can see that the growth factor is approximately 25%.

28. A driver leaves home for a business trip and drives at a constant speed of 60 miles per hour for 2 hours. Her car gets a flat tire, and she spends 30 minutes changing the tire. She resumes driving and drives at 30 miles per hour for the remaining one hour until she reaches her destination.

On the set of axes below, draw a graph that models the driver’s distance from home.

Graph a line with a slope of 60 from time 0 to time 2. Graph a horizontal line (slope 0) from 2 to 2.5. Graph a line with a slope of 30 from time 2.5 to 3.5.

29. How many real solutions does the equation x² - 2x + 5 = 0 have? Justify your answer.

None. If you complete the square, you will get (x - 1)² = -4, which has no real roots.

Or you could find the discriminant (of the quadratic formula):

Alternatively, I don't know if you would get credit for just saying that you graphed y = x² - 2x + 5 and it didn't have any roots, or it didn't cross the x-axis. On the other hand, if you show that the minimum point for the parabola is above the x-axis and, therefore, has no roots, that might have been acceptable.

30. The number of carbon atoms in a fossil is given by the function y = 5100(0.95)^x, where x represents the number of years since being discovered.
What is the percent of change each year? Explain how you arrived at your answer.

There is a 5% decrease each year. The decay factor is .95, and 1.00 - .95 = .05, which is 5%.
If you left it at .05, you probably lost a point because it asked for percent. If you didn't say "decrease", you probably lost a point.

31. A toy rocket is launched from the ground straight upward. The height of the rocket above the ground, in feet, is given by the equation h(t)= 16t² + 64t, where t is the time in seconds.
Determine the domain for this function in the given context. Explain your reasoning.

The domain of the function is 0 < t < 4.
Time cannot be negative. At t=0, h(0) = -16(0)² + 64(0) = 0 + 0 = 0
At t=1, h(1) = -16(1)² + 64(1) = -16 + 64 = 48
At t=1, h(2) = -16(2)² + 64(2) = -64 + 128 = 64
At t=1, h(3) = -16(3)² + 64(3) = -144 + 192 = 48
At t=1, h(4) = -16(4)² + 64(4) = -256 + 256 = 0. The rocket hits the ground.
If t > 4, the rocket would have negative height, which is impossible.

32. Jackson is starting an exercise program. The first day he will spend 30 minutes on a treadmill. He will increase his time on the treadmill by 2 minutes each day. Write an equation for T(d), the time, in minutes, on the treadmill on day d.

Find T(6), the minutes he will spend on the treadmill on day 6.

First part: T(d) = 30 + 2(d - 1), 30 minutes for the first day, plus 2 more each additional day.
Second part: T(6) = 30 + 2(6 - 1) = 30 + 2(5) = 30 + 10 = 40 minutes,