Topics
2. Proportional Relationships
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United States of AmericaVirginia
Grade 7

2.06 Graph proportional relationships

Lesson
Practice

Graph proportional relationships

Recall that a table of values shows the relationship between two quantities (usually represented by x and y). Let's construct our own table of values using the proportional equation:

y = 3x

The table of values for this equation connects the y-value that we get from substituting in a variety of x-values. Let's complete the table of values below:

x-2-101234
y

To substitute x = 1 into the equation y = 3x, we want to replace the variable x with the number 1.

So for x = 1, we get:

\displaystyle y\displaystyle =\displaystyle 3 \cdot 1Substitute x=1
\displaystyle y\displaystyle =\displaystyle 3Evaluate

So we know that 3 is the y-value corresponding to x=1.

x-101234
y3

If we substitute the remaining values of x, we can complete the table.

x-101234
y-3036912
-7
-6
-5
-4
-3
-2
-1
1
2
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5
6
7
x
-7
-6
-5
-4
-3
-2
-1
1
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7
y

\,\\\,Each column in a table of values may be grouped together in the form (x,\,y). We call this pairing an ordered pair.

The table has the following ordered pairs:

\left(-1,\,-3 \right),\,\left(0,\,0 \right),\,\left(1,\,3 \right),\,\left(2,\,6 \right),\,\left(3,\,9 \right),\,\left(4,\,12\right)

We can plot each ordered pair as a point on the coordinate plane.

Notice that our graph is linear and goes through the origin which confirms it is proportional.

-7
-6
-5
-4
-3
-2
-1
1
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x
-7
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y

\,\\\,Now that we have drawn the ordered pairs from the table of values, we can draw the graph that passes through these points.

The slope of this line is 3, and this is the graph of y = 3x which we used to complete the table of values.

We can also graph proportional relationships using the slope and a point on the line.

-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
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6
y

\,\\\,Start by plotting a single point. If the relationship is proportional, we can start at the origin.

We can create a slope triangle from our starting point by using the slope: m = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{3}{1}

Starting at (0,\,0) we move up 3 units and to the right 1 unit before placing another point.

We can repeat this process to get more points.

-4
-3
-2
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1
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x
-3
-2
-1
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y

To find points to the left, we can reverse both directions and go down 3 units from the origin and to the left 1 unit.

This works because \dfrac{-3}{-1}=3 which is the slope.

Now, we can draw a line connecting the points. Notice it is exactly the same line as we graphed using the table.

The slope triangle approach works for various slopes:

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-3
-2
-1
1
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x
-8
-7
-6
-5
-4
-3
-2
-1
1
y
\text{Slope}=-4=\dfrac{-4}{1}: Move 4 units down and 1 unit right
-4
-3
-2
-1
1
2
3
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x
-4
-3
-2
-1
1
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y
\text{Slope}=\dfrac{1}{2}: Move 1 unit up and 2 units right

Notice, in these examples we let the change in x (denominator) be positive. Remember, we can find points in the opposite direction by changing the sign in both the numerator and denominator.

Examples

Example 1

Consider the equation y=-\dfrac{x}{7}.

a

Complete the table of values:

x-7-430
y
Worked Solution
Create a strategy

Substitute each values from the tables into the given equation.

Apply the idea

For x=-7:

\displaystyle y\displaystyle =\displaystyle -\dfrac{-7}{7}Substitute x=-7
\displaystyle =\displaystyle \dfrac{7}{7}Evaluate the adjacent signs
\displaystyle =\displaystyle 1Evaluate

Similarly, if we substitute the other values of x, ( x=-4,\, x=3,\, x=0 ), into y=-\dfrac x 7, we get:

x-7-430
y1\dfrac 47-\dfrac 370
b

Draw the graph of y=-\dfrac{x}{7}.

Worked Solution
Create a strategy

Use the plotted points on the coordinate plane from part (a).

Apply the idea

The equation y=-\dfrac{x}{7} must pass through each of the plotted points.

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x
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y
Reflect and check

We could also graph this relationship using slope triangles.

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x
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y

\,\\\,The equation y = -\dfrac{x}{7} is equivalent to y = -\dfrac{1}{7} x. The slope of the equation is -\dfrac{1}{7}.

m = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{-1}{7}

So the change in y is -1 which means the slope triangle will go down 1. Our change in x is +7 which means our slope triangle will go right 7.

Continue to add points in this way and connect them to form a line.

Example 2

Consider the equation y=4x.

a

Graph the equation on a coordinate plane.

Worked Solution
Create a strategy

Let's use slope triangles to graph the line. We will need to find the change in y and change in x.

Apply the idea

Remember equations for proportional relationships are in the form y = mx where m is the slope. From the equation y = 4x, we know the slope is 4.

m = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{4}{1}

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1
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x
-4
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y

\,\\\,Start by plotting a point at the origin.

From there, move up 4 units and to the right 1 unit.

\,\\\,This triangle will end on point \left(1,\,4 \right). We can repeat this process for multiple points.

Reflect and check

We could also graph by creating a table of values:

x-2-101
y-8-404
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
y

The points from the table have the coordinates (-2,\,-8),\,(-1,\,-4),\,(0,\,0),\,(1,\,4).

Even if we graphed different points, both graphs represent the same line.

b

Is the graph of y=4x linear?

Worked Solution
Create a strategy

Check the graph from part (a) to see if it makes a straight line.

Apply the idea
-6
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1
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x
-8
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y

Since the line formed is a straight line, the relationship linear.

Example 3

Plot the graph of the line whose slope is -3 and passes through the point \left(-2,\,6\right).

Worked Solution
Create a strategy

Use the slope to find another point on the line and then graph it.

Apply the idea

The slope -3 means the y-coordinate will decrease by 3 units if we increase the x-coordinate by 1. Decreasing the y-coordinate by 3 units and then decreasing the x-coordinate by 1 unit, then the new point is (-1,\,3).

Plotting the line whose slope is -3 and passes through the points (-2,\,6) and (-1,\,3), we have

-4
-3
-2
-1
1
2
3
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x
-1
1
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7
y
Reflect and check

We can also create a table of values from the slope and point. Start by adding the point \left(-2,\,6 \right) to the table:

x-2
y6

The slope of -3 tell us every 1 unit change in x results in a -3 unit change in y. We can use this pattern to create a table of equivalent ratios. The ratio y:x is -3:1. So as the x-values are increasing by 1, while the y-values are decreasing by 3. This gives the table:

x-2-1012
y630-3-6
Idea summary

To graph a proportional relationships using table of values:

  1. Complete the table of values by substituting each given x-value into the equation.

  2. Set the x and y-values as ordered pairs (x,\,y) to plot on the graph.

  3. Connect the points with a line.

To graph a proportional relationships using the slope of a line and a point:

  1. Plot any point on the line. You can use the origin since it is proportional.

  2. Find the second point by using a slope triangle. The change in y tells you how far to go up or down. The change in x tells you how far to go right.

  3. Connect the two points through a line.

Outcomes

7.PFA.1

The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in y = mx form, and graphs, including problems in context.

7.PFA.1c

Graph a line representing a proportional relationship, between two quantities given an ordered pair on the line and the slope, m, as rate of change. Slope may include positive or negative values.

7.PFA.1d

Graph a line representing a proportional relationship between two quantities given the equation of the line in the form y = mx, where m represents the slope as rate of change. Slope may include positive or negative values.

7.PFA.1e

Make connections between and among representations of a proportional relationship between two quantities using problems in context, tables, equations, and graphs. Slope may include positive or negative values (slope will be limited to positive values in a contextual situation).

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