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.
The graph of y = (3/2)x^2 is a parabola that opens upwards (since the coefficient of x^2 is positive).
Looking at the options, we can rule out options A and C because the vertex (the lowest point on the parabola) is at x=0, y=0 and the y-coordinates of the other two points are not consistent with the equation.
Option B has the vertex at (0,0), which is consistent with the equation, and passes through the points (-2,4) and (2,4), which are also consistent with the equation.
Option D also has the vertex at (0,0), but passes through the points (-2,6) and (2,6), which are not consistent with the equation.
Therefore, the graph of y = (3/2)x^2 is option B.
Graphing the Coordinate Plane
The table lists the total number of bags of grass seed it took to plant a certain number of lawns. All of the lawns were the same size.
A table titled 'Grass Seed Used' has 6 rows and 2 columns. The first row contains the titles 'Number of Lawns' and 'Number of Bags of Grass Seed Used.' In order under 'Number of Lawns' are the numbers 4, 5, 6, 7, and 8. Under 'Number of Bags' are the numbers 22, 27.5, 33, 38.5, and 44.
Which equation represents the relationship between n, the number of lawns, and b, the number of bags of grass seed?
A. b = 5n
B. b = 5.5n
C. b = n + 18
D. b = n + 22
To find the equation that represents the relationship between the number of lawns (n) and the number of bags of grass seed used (b), we can look for a pattern in the table.
We can see that as the number of lawns increases by 1, the number of bags of grass seed used also increases by the same amount each time. This means that the relationship between n and b is a linear one, where b is the dependent variable and n is the independent variable.
To find the equation, we can use the formula for the equation of a line in slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. In this case, the slope represents the rate of change in the number of bags of grass seed used for each additional lawn. We can find the slope by looking at the change in b over the change in n:
slope = (change in b) / (change in n)
From the table, we can see that:
(change in b) = 5.5 - 22 = 5.5
(change in n) = 5 - 4 = 1
Therefore, the slope is:
slope = (5.5 - 22) / (5 - 4) = -16.5
Using the point-slope form of the equation of a line, we have:
b - 22 = -16.5(n - 4)
Simplifying, we get:
b = -16.5n + 88
Therefore, the equation that represents the relationship between n and b is:
b = -16.5n + 88
So, the answer is none of the provided options.
The table lists the total number of bags of grass seed it took to plant a certain number of lawns. All of the lawns were the same size.
Which equation represents the relationship between n, the number of lawns, and b, the number of bags of grass seed?
A. b = 5n
B. b = 5.5n
C. b = n + 18
D. b = n + 22
To find the equation that represents the relationship between the number of lawns (n) and the number of bags of grass seed used (b), we can look for a pattern in the table.
We can see that as the number of lawns increases by 1, the number of bags of grass seed used also increases by the same amount each time. This means that the relationship between n and b is a linear one, where b is the dependent variable and n is the independent variable.
To find the equation, we can use the formula for the equation of a line in slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. In this case, the slope represents the rate of change in the number of bags of grass seed used for each additional lawn. We can find the slope by looking at the change in b over the change in n:
slope = (change in b) / (change in n)
From the table, we can see that:
(change in b) = 5.5 - 22 = 5.5
(change in n) = 5 - 4 = 1
Therefore, the slope is:
slope = (5.5 - 22) / (5 - 4) = -16.5
Using the point-slope form of the equation of a line, we have:
b - 22 = -16.5(n - 4)
Simplifying, we get:
b = -16.5n + 88
Therefore, the equation that represents the relationship between n and b is:
b = -16.5n + 88
So, the answer is none of the provided options.
Graphing the Coordinate Plane
Note: For questions 19 and 20, your teacher will grade your response to ensure you receive proper credit for your answer.
Point A(4, 2) is translated according to the rule (x, y) right arrow (x + 1, y – 5) and then reflected across the y-axis.
a) In which quadrant of the coordinate plane is point A located?
b) What are the coordinates of translated point A’? In which quadrant of the coordinate plane is point A’ located?
c) What are the coordinates of reflected point A’’? In which quadrant of the coordinate plane is point A’’ located?
a) Point A is located in quadrant I since both x and y coordinates are positive.
b) To apply the translation rule, we add 1 to the x-coordinate of point A and subtract 5 from its y-coordinate.
x-coordinate of A' = x-coordinate of A + 1 = 4 + 1 = 5
y-coordinate of A' = y-coordinate of A - 5 = 2 - 5 = -3
Therefore, the coordinates of translated point A' are (5, -3), which is located in quadrant IV since the x-coordinate is positive and the y-coordinate is negative.
c) To reflect point A' across the y-axis, we change the sign of the x-coordinate while keeping the sign of the y-coordinate the same:
x-coordinate of A'' = -x-coordinate of A' = -5
y-coordinate of A'' = y-coordinate of A' = -3
Therefore, the coordinates of reflected point A'' are (-5, -3), which is located in quadrant III since both the x-coordinate and y-coordinate are negative.
Can the figure below tessellate a plane? Explain your answer.
A shape is formed by cutting out a smaller semicircle from a larger semicircle. The remaining closed figure resembling an arch is shown.
No, the figure cannot tessellate a plane.
In order for a figure to tessellate a plane, copies of the figure must be able to be arranged without any gaps or overlaps to completely cover the plane, like tiles on a floor.
The given figure is not a regular polygon or a shape with straight edges. Instead, it has a curved boundary. No matter how we arrange multiple copies of this figure, the edges will always overlap or leave gaps, and therefore it cannot tessellate a plane.
Graphing the Coordinate Plane Unit Test
Essay
Note: For questions 21–22, your teacher will grade your response to ensure you receive proper credit for your answer.
Look at the shaded triangle in the drawing below. Describe in detail the transformations needed to create the rest of the square design. You can use a combination of translations, reflections, and rotations of the shaded triangle (Figure I) to create the square design.
A square is drawn on a coordinate plane with the center of the square on the origin. Two diagonals are drawn through the origin forming four congruent triangles on the inside of the square. The triangle that overlaps the positive y-axis is shaded.
To create the rest of the square design using transformations of the shaded triangle, we can proceed as follows:
1. Reflect the shaded triangle over the y-axis. This creates another congruent triangle in quadrant II.
2. Translate the original shaded triangle and the triangle created by the reflection one unit to the right. This creates two new triangles in quadrants I and IV.
3. Reflect the two triangles created in step 2 over the x-axis. This creates two more triangles in quadrants II and III.
4. Translate the two triangles created in step 3 one unit down. This creates two more triangles in quadrants I and IV.
5. Reflect the two triangles created in step 4 over the y-axis. This creates two more triangles in quadrants II and III.
At this point, we should have eight congruent triangles in total, forming the complete square design inside the square. We can check that these triangles are all congruent by noting that they were obtained from repeated reflections and translations of the original shaded triangle, which preserves size and shape.
Alternatively, we could have rotated the shaded triangle by 90 degrees (or -90 degrees) around the origin to obtain each new triangle. This is because the diagonals of the square divide it into four congruent right triangles, each of which is related to the others by a rotation of 90 degrees around the
graphing the coordinate plane
The coordinates of an ordered pair have opposite signs. In which quadrant(s) must the ordered pair lie? Explain.