Use the image to answer the question.
An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 1 comma 5 right parenthesis and left parenthesis 2 comma 10 right parenthesis.
The graph shows the proportional relationship. Derive the equation of the line y=mx through the origin.
(1 point)
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To find the equation of the line with a proportional relationship passing through the points (1, 5) and (2, 10), we can use the formula for slope:
slope (m) = (change in y) / (change in x)
Substituting the values from the given points:
m = (10 - 5) / (2 - 1)
m = 5 / 1
m = 5
Since the line passes through the origin (0, 0), the equation of the line y = mx becomes y = 5x.
Graph the proportional relationship y=2x by plotting points.(1 point)
Responses
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 1 comma 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 1 comma 4 right parenthesis.
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 1 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 1 right parenthesis.
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The correct image is:
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: (0, 0), (1, 2), and (2, 4).
Use the image to answer the question.
An illustration shows Quadrant 1 of a coordinate plane. The x-axis is labeled Ounces and ranges from 0 to 15 in 3 unit increments. The y-axis ranges from 0 to 153 in 9 unit increments. The graph is titled Calories per Ounce. A line connects five points plotted on the graph. The coordinates of the plotted points are left parenthesis 0 comma 0 right parenthesis, left parenthesis 3 comma 36 right parenthesis, left parenthesis 6 comma 72 right parenthesis, left parenthesis 9 comma 108 right parenthesis, and left parenthesis 12 comma 144 right parenthesis.
The proportional relationship between calories and ounces of soda is shown in the graph. How many calories are in 1 ounce?
(1 point)
There are
calories in 1 ounce of soda.
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To find the number of calories in 1 ounce of soda, we can use the graph and the given points.
From the graph, we can see that for every 3 ounce increment on the x-axis, the number of calories increases by 36 (from 0 to 36, 36 to 72, 72 to 108, and 108 to 144).
So, the difference in calories for every 3 ounce increment is 36.
To find the number of calories in 1 ounce, we can divide this difference by 3:
36 calories / 3 = 12 calories
Therefore, there are 12 calories in 1 ounce of soda.
Question
Use the image to answer the question.
An illustration shows two graphs depicting earnings per hour for employees 1 and 2. The first graph shows earnings per hour for employee 1. The horizontal axis shows hours ranging from 0 to 8 in increments of 2. The vertical axis shows earnings in dollars ranging from 0 to 80 in increments of 5. Four data points are plotted on the graph at left parenthesis 0 comma 0 right parenthesis, left parenthesis 2 comma 25 right parenthesis, left parenthesis 4 comma 50 right parenthesis, and left parenthesis 6 comma 75 right parenthesis. A line connects all the data points. The second graph shows earnings per hour for employee 2. The horizontal axis shows hours ranging from 0 to 6 in increments of 1. The vertical axis shows earnings in dollars ranging from 0 to 80 in increments of 5. Four data points are plotted on the graph at left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 15 right parenthesis, left parenthesis 3 comma 45 right parenthesis, and left parenthesis 5 comma 75 right parenthesis. A line connects all the data points.
The earnings for Employee 1 and Employee 2 are displayed in the following graphs. Which employee earns more per hour?
(1 point)
Employee
earns more per hour.
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To determine which employee earns more per hour, we can compare the slopes of the lines on the two graphs.
For Employee 1:
Hours: 0, 2, 4, 6
Earnings: 0, 25, 50, 75
Change in earnings: 75 - 0 = 75
Change in hours: 6 - 0 = 6
Slope of Employee 1's line: (change in earnings) / (change in hours) = 75/6 = 12.5
For Employee 2:
Hours: 0, 1, 3, 5
Earnings: 0, 15, 45, 75
Change in earnings: 75 - 0 = 75
Change in hours: 5 - 0 = 5
Slope of Employee 2's line: (change in earnings) / (change in hours) = 75/5 = 15
Comparing the slopes, we can see that the slope for Employee 2 is greater than the slope for Employee 1. This means that Employee 2 earns more per hour than Employee 1.
Use the image to answer the question.
An illustration shows a coordinate plane with the x-axis ranging from negative 9 to 9 in one unit increments, and the y-axis ranging from negative 11 to 11 in one unit increments. A solid line passes through four plotted points and extends beyond. A solid line with arrows at both ends passes through points with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, left parenthesis 2 comma 6 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The solid line forms the hypotenuse for two triangles. The first triangle is marked with a dotted line and has vertices with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 0 right parenthesis, and left parenthesis 1 comma 3 right parenthesis. The base is labeled as 1, and the perpendicular height on the right is labeled as 3. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 1 comma 3 right parenthesis, left parenthesis 3 comma 3 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The base is labeled as 2, and the perpendicular height on the right is labeled as 6.
Use similar triangles to determine the slope of the line.
(1 point)
The slope is
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To determine the slope of the line, we can use the similar triangles formed by the line and the two triangles with known dimensions.
In the first triangle, the base is labeled as 1 and the perpendicular height on the right is labeled as 3.
In the second triangle, the base is labeled as 2 and the perpendicular height on the right is labeled as 6.
Since the two triangles are similar, we can set up a proportion to find the relationship between the base and the perpendicular height:
(base of second triangle) / (perpendicular height of second triangle) = (base of first triangle) / (perpendicular height of first triangle)
2 / 6 = 1 / 3
To simplify, we can cross-multiply:
2 * 3 = 6 * 1
6 = 6
This shows that the two triangles are indeed similar.
Since the line passes through the points (0, 0) and (3, 9), we can use these coordinates to calculate the slope using the formula:
slope = (change in y) / (change in x)
Using the coordinates (0, 0) and (3, 9):
change in y = 9 - 0 = 9
change in x = 3 - 0 = 3
slope = 9 / 3 = 3
Therefore, the slope of the line is 3.
How could you write the equation of a line that would be parallel to y=−3x−2 but not share any points?(1 point)
Responses
y=3x+8
y equals 3 x plus 8
y=3x−2
y equals 3 x minus 2
y=−3x+8
y equals negative 3 x plus 8
y=−3x−2
The correct equation of a line that would be parallel to y = -3x - 2 but not share any points is:
y = -3x + 8
This equation has the same slope (-3) as the original line but a different y-intercept (8 instead of -2), ensuring that it is parallel to the given line but does not intersect with it.
Question
Use the table to answer the question.
($) Cost of Candy Bar 1
Quantity ($) Cost
1 1.25
2 2.50
3 3.75
4 5.00
5 6.25
The cost of Candy Bar 1 is displayed in the table. The proportional relationship between the cost and quantity for Candy Bar 2 is represented by the equation y=x . Which candy bar is the cheapest cost per bar? Enter 1 for Candy Bar 1. Enter 2 for Candy Bar 2.
(1 point)
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To determine which candy bar has the cheapest cost per bar, we need to compare the cost per bar for both candy bars.
For Candy Bar 1:
The cost for 1 bar is $1.25.
Therefore, the cost per bar for Candy Bar 1 is $1.25.
For Candy Bar 2:
The cost per bar is represented by the equation y = x.
When x = 1, y = 1.
Therefore, the cost per bar for Candy Bar 2 is also $1.
Since both candy bars have the same cost per bar of $1, they have the same cheapest cost per bar.
The answer is: Both Candy Bar 1 and Candy Bar 2 have the same cheapest cost per bar.
Use the image to answer the question.
An illustration of a coordinate plane with four quadrants shows the x-axis ranging from negative 9 to 9 and the y-axis ranging from negative 11 to 11 in increments of 1. A solid upward line with arrows at both ends is plotted on the graph. The line passes through the plotted points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 3 right parenthesis.
Use the graph to derive the equation of the line in slope-intercept form.
(1 point)
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To derive the equation of the line in slope-intercept form, we can use the formula:
y = mx + b
where m is the slope of the line and b is the y-intercept.
From the graph, we can see that the line passes through the points (0, -2) and (1, 3).
To find the slope (m), we can use the formula:
m = (change in y) / (change in x)
Substituting the values from the given points:
m = (3 - (-2)) / (1 - 0)
m = 5 / 1
m = 5
Now, we can find the y-intercept (b) by substituting one of the given points and the slope into the equation:
y = mx + b
Using the point (0, -2):
-2 = 5(0) + b
-2 = b
So the y-intercept (b) is -2.
Therefore, the equation of the line in slope-intercept form is:
y = 5x - 2.