Graph Proportional Relationships, Interpreting the Unit Rate as the Slope: CCSS.Math.Content.8.EE.B.5 - 8th Grade Math

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Question

What is the slope of the line that passes through the points and ?

Answer

The slope of a line is sometimes referred to as "rise over run." This is because the formula for slope is the change in y-value (rise) divided by the change in x-value (run). Therefore, if you are given two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$ , the slope of their line can be found using the following formula: $slope=(y_{1}-y_{2})/(x_{1}-x_{2})$

This gives us .

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Question

Given the points and , find the slope of the line.

Answer

The formula for the slope of a line is $\frac{y_2-y_1}{x_2-x_1}$ .

We then plug in the points given: $\frac{9-3}{8-4}=\frac{6}{4}$ which is then reduced to $\frac{3}{2}$ .

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Question

Given points and , what is the slope of the line connecting them?

Answer

Write the slope formula. Plug in the points and solve.

$m= \frac{y_2-y_1}{x_2-x_1}= \frac{3-2}{-1-1}= -\frac{1}{2}$

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Question

What is the slope of the line connecting the points $(2,\sqrt2)$ and ?

Answer

Write the slope formula. Plug in the point, and simplify.

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{2-\sqrt2}{1-2}=\frac{2-\sqrt2}{-1}= -(2-\sqrt2) = \sqrt2-2$

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Question

What is the slope of a line with an -intercept is and another -intercept of ?

Answer

The -intercept is the value when .

Therefore, since the two -intercepts are and , the points are and .

Write the slope formula, plug in the values, and solve.

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-0}{6-4}= \frac{0}{2} =0$

The slope is zero.

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Question

What is the slope m of the line below?

Answer

To answer this question you should realize that the equation for the line is given in point - slope form. The standard point slope form of a line is given below:

m represents the slope of the line so all we have to do is recognize the correct line form and we automatically know that the slope is

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Question

A line crosses the x-axis at and the y-axis at . What is the slope of this line?

Answer

Given the points,

$\(x_1,y_1)=(0,6) \(x_2,y_2)=(6,8)$ .

We compute slope (m) as follows:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-6}{6-0}=\mathbf{\frac{1}{3}}$

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Question

Find the slope of the line that passes through the points:

and

Answer

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Now, substitute in the information using the given points.

$\text{Slope}=\frac{4-2}{-12-12}$

Simplify.

$\text{Slope}=\frac{2}{-24}$

Solve.

$\text{Slope}=-\frac{1}{12}$

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Question

Find the slope of the line that passes through the points:

and

Answer

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Now, substitute in the information using the given points.

$\text{Slope}=\frac{-20-4}{-10-2}$

Simplify.

$\text{Slope}=\frac{-24}{-12}$

Solve.

$\text{Slope}=2$

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Question

Find the slope of the line that passes through the points:

and

Answer

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Now, substitute in the information using the given points.

$\text{Slope}=\frac{1-5}{-3-4}$

Simplify.

$\text{Slope}=\frac{-4}{-7}$

Solve.

$\text{Slope}=\frac{4}{7}$

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Question

Find the slope of the line that passes through the points:

and

Answer

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Now, substitute in the information using the given points.

$\text{Slope}=\frac{13-15}{-39-8}$

Simplify.

$\text{Slope}=\frac{-2}{-47}$

Solve.

$\text{Slope}=\frac{2}{47}$

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Question

Find the slope of the line that passes through the points:

and

Answer

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Now, substitute in the information using the given points.

$\text{Slope}=\frac{6-(-10)}{5-4}$

Simplify.

$\text{Slope}=\frac{16}{1}$

Solve.

$\text{Slope}=16$

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Question

Select the graph that has a unit rate of

Answer

The unit rate of a graph is the slope of the graph. To solve for the slope we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Based on our formula, we first need to find two points for each of the graph, and then we can solve for the slope.

Graph A

Points: and

Slope: $\frac{7-5}{2-1}=\frac{2}{1}=2$

Graph B

Points: and

Slope: $\frac{13-8}{2-1}=\frac{5}{1}=5$

Graph C

Points: and

Slope: $\frac{12-8}{2-1}=\frac{4}{1}=4$

Graph D

Points: and

Slope: $\frac{9-6}{2-1}=\frac{3}{1}=3$

We are looking for the graph with a slope, or unit rate, of ; thus, our correct answer is Graph A.

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Question

Select the graph that has a unit rate of

Answer

The unit rate of a graph is the slope of the graph. To solve for the slope we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Based on our formula, we first need to find two points for each of the graphs, and then we can solve for the slope.

Graph A

Points: and

Slope: $\frac{7-5}{2-1}=\frac{2}{1}=2$

Graph B

Points: and

Slope: $\frac{13-8}{2-1}=\frac{5}{1}=5$

Graph C

Points: and

Slope: $\frac{12-8}{2-1}=\frac{4}{1}=4$

Graph D

Points: and

Slope: $\frac{9-6}{2-1}=\frac{3}{1}=3$

We are looking for the graph with a slope, or unit rate, of ; thus, our correct answer is Graph D.

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Question

Select the graph that has a unit rate of

Answer

The unit rate of a graph is the slope of the graph. To solve for the slope we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Based on our formula, we first need to find two points for each of the graphs, and then we can solve for the slope.

Graph A

Points: and

Slope: $\frac{7-5}{2-1}=\frac{2}{1}=2$

Graph B

Points: and

Slope: $\frac{13-8}{2-1}=\frac{5}{1}=5$

Graph C

Points: and

Slope: $\frac{12-8}{2-1}=\frac{4}{1}=4$

Graph D

Points: and

Slope: $\frac{9-6}{2-1}=\frac{3}{1}=3$

We are looking for the graph with a slope, or unit rate, of ; thus, our correct answer is Graph C.

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Question

Select the graph that has a unit rate of

Answer

The unit rate of a graph is the slope of the graph. To solve for the slope we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Based on our formula, we first need to find two points for each of the graphs, and then we can solve for the slope.

Graph A

Points: and

Slope: $\frac{7-5}{2-1}=\frac{2}{1}=2$

Graph B

Points: and

Slope: $\frac{13-8}{2-1}=\frac{5}{1}=5$

Graph C

Points: and

Slope: $\frac{12-8}{2-1}=\frac{4}{1}=4$

Graph D

Points: and

Slope: $\frac{9-6}{2-1}=\frac{3}{1}=3$

We are looking for the graph with a slope, or unit rate, of ; thus, our correct answer is Graph B.

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Question

Select the graph with the greatest unit rate.

Answer

The highest unit rate means that the slope is the highest. So, first we ned to solve for our slopes.

To solve for the slope we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Based on our formula, we first need to find two points for each of the graphs, and then we can solve for the slope.

Graph A

Points: and

Slope: $\frac{7-5}{2-1}=\frac{2}{1}=2$

Graph B

Points: and

Slope: $\frac{13-8}{2-1}=\frac{5}{1}=5$

Graph C

Points: and

Slope: $\frac{12-8}{2-1}=\frac{4}{1}=4$

Graph D

Points: and

Slope: $\frac{9-6}{2-1}=\frac{3}{1}=3$

We are looking for the graph with the greatest slope, or unit rate; thus, our correct answer is Graph B.

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Question

Select the graph with the lowest unit rate.

Answer

The lowest unit rate means that the slope is the lowest. So, first wee ned to solve for our slopes.

To solve for the slope we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Based on our formula, we first need to find two points for each of the graphs, and then we can solve for the slope.

Graph A

Points: and

Slope: $\frac{7-5}{2-1}=\frac{2}{1}=2$

Graph B

Points: and

Slope: $\frac{13-8}{2-1}=\frac{5}{1}=5$

Graph C

Points: and

Slope: $\frac{12-8}{2-1}=\frac{4}{1}=4$

Graph D

Points: and

Slope: $\frac{9-6}{2-1}=\frac{3}{1}=3$

We are looking for the graph with the lowest slope, or unit rate; thus, our correct answer is Graph A.

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Question

Select the equation that shows the greatest unit rate.

Answer

Unit rate is the slope of an equation. The equations in our answer options are written in slope intercept form:

is always the slope, and is the -intercept.

We are looking for the equation with the greatest unit rate, or slope. Based on our options $\frac{1}{2}$ is the greatest value of ; thus our correct answer is $y=\frac{1}{2}x+4$

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Question

Select the equation that shows the lowest unit rate.

Answer

Unit rate is the slope of an equation. The equations in our answer options are written in slope intercept form:

is always the slope, and is the -intercept.

We are looking for the equation with the lowest unit rate, or slope. Based on our options $\frac{1}{5}$ is the lowest value of ; thus our correct answer is $y=\frac{1}{5}x+7$

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