What follows is a portion of the Common Core Integrate Algebra exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.
June 2016 Algebra Regents, Part III
33. The height, H, in feet, of an object dropped from the top of a building after t seconds is given by H(t) = -16t2 + 144.
How many feet did the object fall between one and two seconds after it was dropped?
Determine, algebraically, how many seconds it will take for the object to reach the ground.
h(1) = -16(1)2 + 144 = 128
h(2) = -16(2)2 + 144 = 80
h(2) - h(1) = 128 - 80 = 48 feet between the 1st and 2nd second.
Solve for h(t) = 0.
-16(t2 - 9) = 0
-16(t + 3)(t - 3) = 0
t = -3 or t = 3
It takes 3 seconds to reach the ground.
34. The sum of two numbers, x and y, is more than 8. When you double x and add it to y, the sum is
less than 14.
Graph the inequalities that represent this scenario on the set of axes below.
Kai says that the point (6,2) is a solution to this system. Determine if he is correct and explain your reasoning.
The first inequality you need to graph is x + y > 8. The second one is 2x + y < 16.
The graph looks like this:
(graph will be uploaded later)
Looking at the graph, Kai is incorrect because (6, 2) is on a broken line which is not part of the solution set.
Note that if you drew the graph with solid lines, you lost a point for that. However, Kai would have been correct according to that mistaken graph. You have to be consistent.
35. An airplane leaves New York City and heads toward Los Angeles. As it climbs, the plane gradually increases its speed until it reaches cruising altitude, at which time it maintains a constant speed for several hours as long as it stays at cruising altitude. After flying for 32 minutes, the plane reaches cruising altitude and has flown 192 miles. After flying for a total of 92 minutes, the plane has flown a total of 762 miles.
Determine the speed of the plane, at cruising altitude, in miles per minute.
Write an equation to represent the number of miles the plane has flown, y, during x minutes at cruising altitude, only.
Assuming that the plane maintains its speed at cruising altitude, determine the total number of miles the plane has flown 2 hours into the flight.
There was much discussion over this question, but in the end, there was no arguing with the rubric for scoring this question. If you feel it's an unfair question, appeal to the state. In the meantime...
To find the speed at cruising altitude, use the two points given (32 minutes, 192 miles) and (92 minutes, 762 miles).
Speed in Miles per minute = changes in distance (miles) / change in time (minutes)
(762 - 192) / (92 - 32) = 9.5
If you showed your work, you have 1 point already.
The second part was where many students got caught up.
The word only was meant to apply to both the x and the y values, not just the x. In other words, you did not need to account for the distance traveled before reaching cruising altitude.
Because of this, the correct equation was y = 9.5x.
If you included "+ 192", you didn't get credit.
For the last part, you not only need to remember the initial 192 miles, but also the first 32 minutes of the flight. The question states that it is 2 hours into the flight, not 2 hours at cruising altitude. Also remember that you are dealing with miles per minute, so you need to convert.
2 hours = 120 minutes
120 - 32 = 88 minutes at cruising altitude
y = 9.5(88) + 192 = 836 + 192 = 1028 miles
36. On the set of axes below, graph:
How many values of x satisfy the equation f(x) = g(x)? Explain your answer, using evidence from your graphs.
(Graph will be posted later)
It is important that you had a break in the line at x = -1. The linear portion ends, the quadratic portion needed to have an open circle.
Most of the mistakes I saw fit into these categories:
- Draw three equations across the entire plane
- Connecting the two parts of the piecewise function
- Forgetting the open circle
- Shading the graph like it was an inequality
- Graphing a broken line for the quadratic because of the greater than symbol
- Saying that there were no solutions because the three lines didn't intersect at a single point (they didn't have to)
There was one solution because f(x) and g(x) only intersect one time.
You did not have to give the coordinates of the solution, and the solution was NOT a proper explanation. Seriously. You had to reference the fact that the lines only cross/intersect one time so there is one solution.
Also, if you had a graphing error, your final answer had to match the graph you drew. If, for example, you graph g(x) = -1/2x + 1, that would be one graphing error, but that line would intersect f(x) two times. Your answer had to match your graph.
June 2016 Algebra Regents, Part IV
37. Franco and Caryl went to a bakery to buy desserts. Franco bought 3 packages of cupcakes and 2 packages of brownies for $19. Caryl bought 2 packages of cupcakes and 4 packages of brownies for $24. Let x equal the price of one package of cupcakes and y equal the price of one package of brownies.
Write a system of equations that describes the given situation.
On the set of axes below, graph the system of equations.
Determine the exact cost of one package of cupcakes and the exact cost of one package of brownies in dollars and cents. Justify your solution.
The two equations were
2x + 4y = 24
Note: if you used c and b instead of x and y, you lost 1 point because the instructions said to use x and y.
(The graph will be loaded later.)
You could solve the system of equations using elimination. You could also solve them using the functions on the graphing calculator, but you needed to explain how you got your answer. A correct pair of answers without an explanation or procedure was only 1 point instead of 2.
Multiply the first equation by 2 and subtract
2x + 4y = 24
4x = 14
x = 3.50
2(3.50) + 4y = 24
7.00 + 4y = 24
4y = 17y = 4.25