Graphing the Coordinate Plane
The coordinates of three vertices of a rectangle are shown on the grid.
Which could be the coordinates of the fourth vertex?
A. (3, –2)
B. (3, –1)
C. (–2, 3)
D. (4, –2)
To be a rectangle, the opposite sides must be parallel and equal in length. Therefore, the fourth vertex must be the same distance away from one of the given vertices as the other given vertex. Looking at the options, only option A has the same distance from both (2 units). Therefore, the answer is A. (3, –2)
Graphing the Coordinate Plane
Which graph shows the image of a figure after a rotation of 180° around the origin?
A. graph AThe coordinates of the vertices for the first square are upper A superscript 1 baseline left-parenthesis negative 2 comma 3 right-parenthesis, upper B superscript 1 baseline left-parenthesis negative 3 comma 1 right-parenthesis, upper C superscript 1 baseline left-parenthesis negative 1 comma 0 right-parenthesis, and upper D superscript 1 baseline left-parenthesis 2 comma 0 right-parenthesis. The coordinates of the vertices for the second square are upper A left-parenthesis 2 comma 3 right-parenthesis, upper B left-parenthesis 3 comma 1 right-parenthesis, upper C left-parenthesis 1 comma 0 right-parenthesis, and upper D left-parenthesis 2 comma 0 right-parenthesis.
B. graph BThe coordinates of the vertices for the first square are upper A superscript 1 baseline left-parenthesis negative 2 comma 3 right-parenthesis, upper B superscript 1 baseline left-parenthesis negative 3 comma 1 right-parenthesis, upper C superscript 1 baseline left-parenthesis negative 1 comma 0 right-parenthesis, and upper D superscript 1 baseline left-parenthesis 2 comma 0 right-parenthesis. The coordinates of the vertices for the second square are upper A left-parenthesis negative 2 comma 3 right-parenthesis, upper B left-parenthesis 1 comma negative 1 right-parenthesis, upper C left-parenthesis 1 comma 0 right-parenthesis, and upper D left-parenthesis 0 comma negative 2 right-parenthesis.
C. graph CThe coordinates of the vertices for the first square are upper A superscript 1 baseline left-parenthesis negative 2 comma 3 right-parenthesis, upper B superscript 1 baseline left-parenthesis negative 3 comma 1 right-parenthesis, upper C superscript 1 baseline left-parenthesis negative 1 comma 0 right-parenthesis, and upper D superscript 1 baseline left-parenthesis 2 comma 0 right-parenthesis. The coordinates of the vertices for the second square are upper A left-parenthesis negative 5 comma negative 2 right-parenthesis, upper B left-parenthesis negative 4 comma negative 4 right-parenthesis, upper C left-parenthesis negative 2 comma negative 3 right-parenthesis, and upper D left-parenthesis negative 3 comma negative 1 right-parenthesis.
D. graph D
A rotation of 180° around the origin involves flipping the figure over the x-axis and then over the y-axis (or vice versa), which means that each point will have the opposite sign for both x and y coordinates. Looking at the options, only graph B has the image of the original figure after such a rotation. Therefore, the answer is B.
Graphing the Coordinate Plane
Which table shows a proportional relationship?
A. table aIn column 1, x is 1 and y is 2.
In column 2, x is 3 and y is 4.
In column 3, x is 4 and y is 6.
In column 4, x is 6 and y is 7.
B. table bIn column 1, x is 1 and y is negative 2. In column 2, x is 2 and y is 0. In column 3, x is 5 and y is 6. In column 4, x is 8 and y is 12.
C. table cIn column 1, x is 2 and y is negative 4.
In column 2, x is 3 and y is negative 6.
In column 3, x is 5 and y is negative 10.
In column 4, x is 6 and y is negative 12.
D. table DIn column 1, x is 2 and y is 2.
In column 2, x is 4 and y is 3.
In column 3, x is 6 and y is 4.
In column 4, x is 8 and y is 5.
A proportional relationship is one where the ratio between the two variables (y/x) is always the same. Looking at the options, only table A has a constant ratio for each pair of corresponding values: 2/1 = 4/3 = 6/4 = 7/6. Therefore, the answer is A.
Identify the unit rate shown in the graph.
A graph is shown. The horizontal axis is labeled 'Number of Hours' and values range from 0 to 6. The vertical axis is labeled 'Amount Paid' and values range from 0 to 60. A linear graph begins at 0 comma 0 and extends up and to the right to approximately 6 comma 53.
A. $7 per hour
B. $9 per hour
C. 7 hours per $1
D. 9 hours per $1
15 / 22
To find the unit rate, we need to determine the amount paid per hour. Looking at the graph, we can see that at 1 hour, the amount paid is approximately $9, and at 2 hours, the amount paid is approximately $18. This means that for every hour, an additional $9 is paid. Therefore, the unit rate shown in the graph is $9 per hour.
So, the answer is B.
Graphing the Coordinate Plane
The line graphed below shows the rate at which a machine can package pieces of candy.
A coordinate graph is shown.
The graph is titled 'Packaging Candy' and has an x axis labeled 'Number of Minutes' with coordinates labeled from 0 to 15 in multiples of 5. The y axis is labeled 'Pieces of Candy' and is marked from 0 to 3000 in multiples of 1000. A red ray on the graph begins at 0 comma 0 and extends up and to the right through the point 10 comma 2000.
Which gives the slope of the line?
A. 5000 minutes per piece of candy
B. 5000 pieces of candy per minute
C. 200 minutes per piece of candy
D. 200 pieces of candy per minute
The slope of a line is the rate of change of y with respect to x, which tells us how much y changes for each unit change in x. In this case, the units for x are minutes and the units for y are pieces of candy.
Looking at the graph, we can see that the red ray goes through the points (0,0) and (10,2000), which means that in 10 minutes, the machine packaged 2000 pieces of candy. Therefore, the slope of the line is:
(change in y) / (change in x) = (2000 - 0) / (10 - 0) = 200 pieces of candy per minute
So, the answer is D.
Graphing the Coordinate Plane
Which of the following is a graph of y = three-halvesx2?
A. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 2 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 2 right parenthesis.
B. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 4 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
C. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 8 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 8 right parenthesis.
D. This is a graph of a parabola that passes through the points left parenthesis negative 2 comma 6 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.