Graph Proportional Relationships: Unit Rate as Slope | Grade 8 Math | LevelKite

In the last lesson you wrote equations for proportional relationships. In a proportional relationship, $y = kx$ for one fixed number $k$ , called the constant of proportionality. So $y \div x = k$ for every pair with $x \neq 0$ . Now you will graph these relationships and find $k$ on the graph.

Start with a pump that fills an empty tank at a steady unit rate of $3$ liters per minute. A unit rate is the amount of one quantity for exactly $1$ unit of the other. Here that is the water added in $1$ minute. Let $x$ be the time in minutes since the pump started, so $x \ge 0$ . Let $y$ be the water in the tank in liters. Then $y = 3x$ . The graph of this relationship shows each pair $(x, y)$ that fits it as a point on the coordinate plane.

Worked example

Graph the pump equation

Graph $y = 3x$ , where $y$ is the liters of water in the tank after $x$ minutes. Use the whole-number times from $0$ to $4$ minutes.

The equation says the water is 3 times the time, so multiply each time by 3. Start with 0 minutes. Each time and its amount of water form a pair that makes y = 3x true.
$x = 0,\ 1,\ 2,\ 3,\ 4 \quad\Rightarrow\quad y = 0,\ 3,\ 6,\ 9,\ 12$
Write each pair as a point (x, y). Time comes first because it goes on the horizontal axis.
$(0, 0),\ (1, 3),\ (2, 6),\ (3, 9),\ (4, 12)$
Plot each point. Start at the origin, the point (0, 0) where the axes meet. Move right x units and up y units, because the first coordinate gives the distance across and the second gives the distance up. For (2, 6), move right 2 and up 6.
Draw one straight line through the points, starting at the origin. A line fits because each extra minute adds the same 3 liters, so the points climb in equal steps. Times between whole minutes follow y = 3x too, so they lie on the same line.
$x = 1.5:\quad y = 3 \times 1.5 = 4.5$

Answer

The graph of $y = 3x$ for $x \ge 0$ is the straight line that starts at the origin and passes through $(1, 3)$ , $(2, 6)$ , $(3, 9)$ and $(4, 12)$ .

The points

(0, 0)

,

(1, 3)

,

(2, 6)

,

(3, 9)

and

(4, 12)

from the steps all lie on one line that starts at the origin: the graph of

y = 3x

.

In the figure, the horizontal axis shows the time $x$ in minutes. The vertical axis shows the water $y$ in liters. Grid lines are $1$ unit apart on both axes. Each labelled point is one pair from the steps. The line also includes the in-between times, such as the point $(1.5, 4.5)$ from the last step. The line starts at the origin because the tank is empty at the start: $y = 3 \times 0 = 0$ . Every proportional relationship $y = kx$ passes through the origin, since $k \times 0 = 0$ . Here you draw only the part with $x \ge 0$ , because time cannot be negative.

Three places to find the rate of 3 liters per minute for y = 3x

Time x (minutes)	Water y (liters)	Quotient y ÷ x	Change in y when x grows by 1
$0$	$0$	undefined, because $x = 0$	none (first row)
$1$	$3$	$3 \div 1 = 3$	$3 - 0 = 3$
$2$	$6$	$6 \div 2 = 3$	$6 - 3 = 3$
$3$	$9$	$9 \div 3 = 3$	$9 - 6 = 3$
$4$	$12$	$12 \div 4 = 3$	$12 - 9 = 3$

All three readings in the table give $3$ . The water at $x = 1$ is $3$ liters. Each extra minute adds $3$ liters. And $y \div x = 3$ at every point except the origin, where $0 \div 0$ is undefined. That $3$ is the unit rate, $3$ liters per minute. On the graph it shows up as steepness: the line rises $3$ units for each $1$ unit you move right.

The slope of a line measures that steepness. Between two points on the line, the rise is the change in $y$ and the run is the change in $x$ . The slope is $\text{rise} \div \text{run}$ . Take the points $(1, 3)$ and $(4, 12)$ . The rise is $12 - 3 = 9$ liters and the run is $4 - 1 = 3$ minutes. So the slope is $9 \div 3 = 3$ liters per minute, the unit rate again. The run must not be $0$ . A vertical line has run $0$ , and division by $0$ is undefined. So a vertical line has undefined slope and is never the graph of $y = kx$ .

Key concept

The unit rate is the slope

For a proportional relationship $y = kx$ , the graph is a straight line through the origin. Its slope is the unit rate $k$ , in units of $y$ per unit of $x$ . You can find $k$ three ways:

the $y$ -value at $x = 1$ , because $k \times 1 = k$ ;
the rise for each increase of $1$ in $x$ , because $k(x + 1) - kx = k$ ;
the quotient $y \div x$ at any point with $x \neq 0$ , because $kx \div x = k$ .

Take any two points with $x_1 \neq x_2$ . The rise is $k$ times the run, so the slope is $k$ .

\frac{\text{rise}}{\text{run}} = \frac{kx_2 - kx_1}{x_2 - x_1} = \frac{k(x_2 - x_1)}{x_2 - x_1} = k \quad (x_2 \neq x_1)

Now go the other way and read the unit rate from a graph. At a market, the cost of apples is proportional to their weight. Let $x$ be the weight in pounds, with $x \ge 0$ . Let $y$ be the cost in dollars. In the next figure, the horizontal axis shows weight in pounds and the vertical axis shows cost in dollars. Grid lines are $1$ unit apart. Work from the exact coordinates of the labelled points, not from how steep the line looks. The look depends on how the axes are stretched.

The cost line passes through the labelled points

(0, 0)

,

(2, 3)

and

(4, 6)

:

2

pounds cost

3

dollars and

4

pounds cost

6

dollars.

Worked example

Read the unit rate from a graph

The cost of apples is proportional to their weight. Its graph is the straight line through $(0, 0)$ , $(2, 3)$ and $(4, 6)$ , where $x$ is the weight in pounds and $y$ is the cost in dollars. Find the slope, explain what it means, and write the equation.

Find the slope from (2, 3) to (4, 6). The rise is the change in cost, in dollars. The run is the change in weight, in pounds. So the slope is in dollars per pound. The run is 2, not 0, so the division is defined.
$\frac{\text{rise}}{\text{run}} = \frac{6 - 3}{4 - 2} = \frac{3}{2} = 1.5$
Check with the quotient y ÷ x at the two labelled points other than the origin, where x is not 0. Both quotients equal the slope, as they must for a proportional relationship.
$\frac{3}{2} = 1.5, \qquad \frac{6}{4} = 1.5$
So k = 1.5 and the equation is y = 1.5x. It fits all three labelled points. Only one straight line passes through two different points, so the drawn line is the graph of y = 1.5x.
$1.5 \times 0 = 0, \quad 1.5 \times 2 = 3, \quad 1.5 \times 4 = 6$
Interpret the slope. At x = 1 the cost is 1.5 dollars. Each extra pound adds 1.5 dollars, which is the rise of 1.5 for each 1 to the right. The point (1, 1.5) sits halfway between grid lines, so you compute its height instead of reading it off the picture.
$1.5 \times 1 = 1.5$

Answer

The slope is $1.5$ , so the apples cost $1.50$ dollars per pound, and the equation is $y = 1.5x$ .

A straight line is not always a proportional relationship. Return to the pump, but suppose the tank already holds $1$ liter when the pump starts. Then $y = 3x + 1$ . The axes measure the same quantities as before: time $x$ in minutes across and water $y$ in liters up. Grid lines are $1$ unit apart. In the figure below, Line A is $y = 3x$ and Line B is $y = 3x + 1$ . Line B rises $3$ for each $1$ to the right, like Line A, because the pump still adds $3$ liters each minute. But at $x = 0$ , Line B gives $y = 3 \times 0 + 1 = 1$ . So it passes through $(0, 1)$ and misses the origin. The two lines look close near the start, but the coordinates show the difference. A proportional relationship must give $y = 0$ when $x = 0$ , so $y = 3x + 1$ is not proportional.

Line A,

y = 3x

, passes through the origin. Line B,

y = 3x + 1

, passes through

(0, 1)

,

(1, 4)

and

(2, 7)

, so it misses the origin.

Takeaway: the graph of a proportional relationship $y = kx$ is a straight line through the origin, and its slope is the unit rate $k$ . You can read $k$ as the $y$ -value at $x = 1$ or as the rise for each increase of $1$ in $x$ . You can also read it as $y \div x$ at any point with $x \neq 0$ . A straight line that misses the origin is not proportional, so check the origin first. Next, you will compare proportional relationships shown in different ways and use them to solve word problems.