The following are some of the multiple questions from the January 2020 New York State Common Core Algebra I Regents exam.
January 2020 Algebra I, Part III
Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.
33. Michael threw a ball into the air from the top of a building. The height of the ball, in feet, is modeled by the equation h = -16t^{2} + 64t + 60, where t is the elapsed time, in seconds. Graph this equation on the set of axes below.
Determine the average rate of change, in feet per second, from where Michael released the ball to when the ball reached its maximum height.
Answer:
Enter the equation into your graphing calculator and look at the table. You will see that there are points at (0,60), (1,108), (2,124), (3,108), (4,60) and (5,-20).
If you find the Zero, or just check a few numbers, you will see that the ball will hit the ground between 4.7 and 4.8 seconds.
To find the average rate of change from when he released the ball to its maximum height, find the slope of the line between (0,60) and (2,124)
(124 - 60)/(2 - 0) = 64/2 = 32 ft per second.
34. Graph the system of inequalities
3x + 4y + 4 > 0
Stephen says the point (0, 0) is a solution to this system. Determine if he is correct, and explain your reasoning.
Answer:
First thing you probably want to do is to rewrite the inequalities in either Standard or Slope-Intercept form.
-x + 2y - 4 < 0
-x + 2y < 4 2y < x + 4 y < 1/2 x + 2 |
3x + 4y + 4 > 0
3x + 4y > -4 4y > -3x - 4 y > -3/4 x - 1 |
Notice that the coefficients for y in each inequality are positive, so the inequality symbols will not flip when rewriting them.
The first inequality is less than, so it will have a broken line and be shaded below.
The second inequality is greater than or equal to, so it will have a solid line and be shaded above.
In the first inequality, the y-intercept is 2 and the slope is 1/2, which you can use to plot other points on this boundary line.
In the second inequality, the y-intercept is -1 and the slope is -3/4, which you can use to plot other points on the solid line.
After you shade both solutions, label the area on the right side, where the two solutions overlap, with a big "S". All the points in this section, including those on the solid line, but not those on the broken line, are solutions to the system of inequalities.
Your graph should look like the following:
Stephen is correct that (0,0) is a solution to the system because (0,0) is in the solution area for both inequalities.
35. The following table represents a sample of sale prices, in thousands of dollars, and number of new homes available at the price in 2017.
State the linear regression function, f(p), that estimates the number of new homes available at a specific sale price, p. Round all values to the nearest hundredth.
State the correlation coefficient of the data to the nearest hundredth. Explain what this means in the context of the problem.
Answer:
Put the first row of numbers into List 1 (L_{1}) in your calculator. Put the second row of numbers into List 2 (L_{2}) in your calculator. Do a linear regression.
You should get, to the nearest hundredth, a=-.79, b=249.86 and r=-.95
The function, f(p) = -.79p + 249.86.
The correlation coefficient is -.95. This is a strong negative correlation. The higher the prices of the homes, the fewer homes are available.
36. The length of a rectangular sign is 6 inches more than half its width. The area of this sign is 432 square inches. Write an equation in one variable that could be used to find the number of inches in the dimensions of this sign.
Solve this equation algebraically to determine the dimensions of this sign, in inches.
Answer:
The length is six inches more than half the width.
So L = 1/2 W + 6
The area is 432 square inches, so (L)(W)=432.
Combine them into one equation and you get (1/2W + 6)(W) = 432
or 1/2W^{2} + 6W = 432
To solve this equation, set the right side equal to 0 and factor.
1/2W^{2} + 6W = 432
1/2W^{2} + 6W - 432 = 0
W^{2} + 12W - 864 = 0
Factors of 864: 2 X 432, 3 X 288, 4 X 216, 6 X 144, 8 X 108, 9 X 96, 12 X 72, 16 X 52, 18 X 48, 24 X 36
(W - 24)(w + 36) = 0
W = 24 or W = -36
Reject the negative answer.
The Width is 24, so the length is 1/2 (24) + 6 = 18.
If you didn't want to work out the factors, you could have used the Quadratic Formula to find the same results. Or Completing the Square, which would have been the easiest in this case because you would have had to find the square root of 900, which is 30.
W^{2} + 12W - 864 = 0
W^{2} + 12W = 864
W^{2} + 12W + 36 = 900
(W + 6)^{2} = 900
W + 6 = SQRT(900) = + 30
W = 30 - 6 = 24 or W = -30 - 6 = -36 (reject the negative number)
January 2020 Algebra I, Part IV
A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.
37. Two families went to Rollercoaster World. The Brown family paid $170 for 3 children and 2 adults. The Peckham family paid $360 for 4 children and 6 adults.
If x is the price of a child's ticket in dollars and y is the price of an adult's ticket in dollars, write a system of equations that models this situation.
State the coordinates of the point of intersection.
Explain what each coordinate of the point of intersection means in the context of the problem.
Answer:
Create a system of equations from the information that you were given.
3x + 2y = 170
4x + 6y = 360
If you can graph those in Standard Form by finding the y-intercept and the slope, GREAT! Otherwise, rewrite them in slope-intercept form, which will allow you to use your graphing calculator to get the table.
3x + 2y = 170
2y = -3x + 170 y = -3/2 x + 85 |
4x + 6y = 360
6y = -4x + 360 y = -4/6 x + 60 y = -2/3 x + 60 |
Plot a point at (0, 85) and use the slope (-3/2) to plot the rest of that line.
Plot a point at (0, 60) and use the slope (-2/3) to plot the rest of that line.
Label the lines! Either write Brown family and Peckham family next to the corresponding line, or write the original equation next to the line. It is important that you state which line is which.
The solution to this system of equations is the point of intersection. The coordinates of the point of intersection are (30, 40).
In the context of this problem, it means that the price of children's tickets are $30 and the price of adult tickets is $40.
End of Exam
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