Unit 3 Linear Relationships — Unit Plan

Graphing is a way to help make sense of relationships.

But the graph of a line on a coordinate plane without labels or a scale isn’t very helpful. Without labels, we can’t tell what the graph is about or what units are being used. Without an appropriate scale, we can’t tell any specific values.

Here are the same graphs, but now with labels and a scale:​​​​​

Notice how adding labels lets us know that the relationship compares time and distance and helps to understand both the speed and pace of two different items. When adding labels to axes, be sure to include units, such as minutes and miles.

Notice how adding a scale makes it possible to identify specific points and values. When adding a scale to an axis, be sure that the space between each grid line represents the same amount.

The scales we choose when graphing a relationship often depend on what information we want to know. For example, consider two water tanks filled at different constant rates.

The relationship between time in minutes $t$ and volume in liters $v$ of Tank A can be described by the equation $v=2.2t$.

For Tank B the relationship can be described by the equation $v=2.75t$

These equations tell us that Tank A is being filled at a constant rate of 2.2 liters per minute and Tank B is being filled at a constant rate of 2.75 liters per minute.

If we want to use graphs to see at what times the two tanks will have 110 liters of water, then using an axis scale from 0 to 10, as shown here, isn't very helpful.

graph, horizontal axis, time in minutes, scale 0 to 9, by 1's. vertical axis, volume in liters, 0 to 9, by 1's. lines, first line passing through origin and 2 comma 5 and 5 tenths. second line passing through origin and 2 comma 4 and 5 tenths.

graph, horizontal axis, time in minutes, scale 0 to 90, by 10's. vertical axis, volume in liters, 0 to 140, by 10's. 2 lines, first line passing through origin and 40 comma 110. second line passing through origin and 50 comma 110.

If we use a vertical scale that goes to 150 liters, a bit beyond the 110 we are looking for, and a horizontal scale that goes to 100 minutes, we get a much more useful set of axes for answering our question.

Now we can see that the two tanks will reach 110 liters 10 minutes apart—Tank B after 40 minutes of filling and Tank A after 50 minutes of filling.

It is important to note that both of these graphs are correct, but one uses a range of values that helps answer the question. In order to always pick a helpful scale, we should consider the situation and the questions asked about it.

When two proportional relationships are represented in different ways, we can compare them by finding a common piece of information.

If we want to know who makes more per hour, we can look at the rate of change for each situation.

In Clare’s equation, we see that the rate of change is 14.50. This tells us that she earns $14.50 per hour. For Jada, we can calculate the rate of change by dividing her earnings in one row by the hours worked in the same row. For example, using the last row, the rate of change is 13.25 since $490.25\div37=13.25$. This tells us that Clare earns $1.25$ more dollars per hour than Jada.

A linear relationship is any relationship between two quantities where one quantity has a constant rate of change with respect to the other. For example, Andre babysits and charges a fee for traveling to and from the job, and then a set amount for every additional hour he works. Since the total amount he charges with respect to the number of hours he works changes at a constant rate, this is a linear relationship. But since Andre charges a fee for traveling, and the graph does not go through the point $(0,0)$, this is not a proportional relationship. Here is a graph of how much Andre charges based on how many hours he works.

Graph, horizontal axis, time in hours, scale 0 to 9, by 1's. vertical axis, amount earned in dollars, scale 0 to 140, by 20's. line starting at 0 comma 10, passing through 2 comma 40 and 60 comma 100.

The rate of change can be calculated using the graph. Since the rate of change is constant, we can take any two points on the graph and divide the amount of vertical change by the amount of horizontal change. For example, the points $(2,40)$ and $(6,100)$ mean that Andre earns 40 dollars for working 2 hours and 100 dollars for working 6 hours. The rate of change is $\frac{100-40}{6-2}=15$ dollars per hour. Andre's earnings go up 15 dollars for each hour of babysitting.

Notice that this is the same way we calculate the slope of the line. That's why the graph is a line and why we call this a “linear relationship.” The rate of change of a linear relationship is the same as the slope of its graph.

Lines drawn on a coordinate plane have a slope and a vertical intercept. The vertical intercept indicates where the graph of the line meets the vertical axis. Since the vertical axis is often referred to as the $y$-axis, the vertical intercept is often called the “$y$-intercept.” A line represents a proportional relationship when the vertical intercept is 0.

Here is a graph of a line showing the amount of money paid for a new cell phone and monthly plan.

The vertical intercept for the graph is at the point $(0,200)$ and means the initial cost for the phone was $200.

A slope triangle connecting the two points $(0,200)$ and $(2,300)$ can be used to calculate the slope of this line. The slope of 50 means that the phone service costs $50 per month in addition to the initial $200 for the phone.

During an early winter storm, snow falls at a rate of $\frac12$ inch per hour. The rate of change, $\frac12$, can be seen in both the equation $y=\frac12x$ and in the slope of the line representing this storm.

The time since the beginning of the storm and the depth of the snow is a linear relationship. This is also a proportional relationship since the depth of snow is 0 inches at the beginning of the storm.

Graph of line. Horizontal axis, time since beginning of storm in hours, scale 0 to 6, by 1’s. Vertical axis, depth of snow in inches, scale 0 to 9, by 1’s. Points on line include 0 comma 0, 2 comma 1 and 4 comma 2.

During a mid-winter storm, snow again falls at a rate of $\frac12$ inch per hour, but this time there were already 5 inches of snow on the ground.

The rate of change, $\frac12$, can still be seen in both the equation and in the slope of the line representing this second storm. 

The 5 inches of snow that were already on the ground can be graphed by translating the graph of the first storm up 5 inches, resulting in a vertical intercept at $(0,5)$. It can also be seen in the equation $y=\frac12x+5$.

This second storm is also a linear relationship, but unlike the first storm, is not a proportional relationship since its graph has a vertical intercept of 5.

At the end of winter in Maine, the snow on the ground was 30 inches deep. Then there was a particularly warm day and the snow melted at the rate of 1 inch per hour. The graph shows the relationship between the time since the snow started to melt and the depth of the remaining snow.

Graphs with a negative slope often describe situations where some quantity is decreasing over time.

Since the depth of the snow decreases by 1 inch per hour, the rate of change is -1 inch per hour and the slope of this graph is -1. The vertical intercept is 30 since the snow was 30 inches high before it started to melt.

graph on grid, origin O. horizontal axis, time since snow started to melt in hours, scale 0 to 11, by 1's. vertical axis, depth of snow in inches, scale 0 to 30, by 5's. line with negative slope drawn that goes through points 2 comma 28, 3 comma 27. triangle with down 1 right 1 drawn below these 2 points. line also goes through 5 comma 25 and 10 comma 20. triangle with down 5 right 5 drawn below these 2 points.

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Graphs with a slope of 0 describe situations where there is no change in the $y$-value even though the $x$-value is changing.

graph on a grid, origin O. horizontal axis, time since winning contest in days, scale 0 to 11, by 1's. vertical axis, balance on fare card in dollars, 0 to 6, by 1's. horizontal line through 2 days comma 5 dollars and 5 days comma 5 dollars. 

For example, Elena wins a prize that gives her free bus rides for a year. Her fare card already had $5 on it when she won the prize. Here is a graph of the amount of money on her fare card after winning the prize. Since she doesn’t need to add or use money from her fare card for the next year, the amount on her fare card will not change. The rate of change is 0 dollars per day and the slope of this graph is 0. All graphs of linear relationships with slopes of 0 are horizontal.

One way to calculate the slope of a line is by drawing a slope triangle. For example, using this slope triangle, the slope of the line is $\text-\frac24$, or $\text-\frac12$. The slope is negative because the line is decreasing from left to right.

Another way to calculate the slope of this line uses just the points $A:(1,5)$ and $B:(5,3)$. The slope is the vertical change divided by the horizontal change, or the change in the $y$-values divided by the change in the $x$-values. Between points $A$ and $B$, the $y$-value change is $3-5=\text-2$ and the $x$-value change is $5-1=4$. This means the slope is $\text-\frac24$, or $\text-\frac12$, which is the same value as the slope calculated using a slope triangle.

Notice that in each of the calculations, the value from point $A$ was subtracted from the value from point $B$. If it had been done the other way around, then the $y$-value change would have been $5-3=2$ and the $x$-value change would have been $1-5=\text-4$, which still gives a slope of $\text-\frac12$. 

A solution to an equation with two variables is any pair of values for the variables that make the equation true. For example, the equation $2x+2y=8$ represents the relationship between the width $x$ and length $y$ for rectangles with a perimeter of 8 units. One solution to the equation $2x+2y=8$ is that the width and length could be 1 and 3, since $2\boldcdot1+2\boldcdot3=8$. Another solution is that the width and length could be 2.75 and 1.25, since $2\boldcdot(2.75)+2\boldcdot(1.25)=8$. There are many other possible pairs of width and length that make the equation true.

The pairs of numbers that are solutions to an equation can be seen as points on the coordinate plane where every point represents a different rectangle whose perimeter is 8 units. Here is part of the line created by all the points $(x,y)$ that are solutions to $2x+2y=8$. In this situation, it makes sense for the graph to only include positive values for $x$ and $y$ since there is no such thing as a rectangle with a negative side length.

Graph of a line, origin O, with grid. Horizontal axis, scale 0 to 5, by 1’s. Vertical axis, scale 0 to 5, by 1’s. Line begins on vertical axis above origin, passes through 1 comma 3 and 2 and 75 hundredths comma 1 and 25 hundredths.

Consider the graph of the  linear equation $2x-4y=12$. 

Since $(0,\text-3)$ is a point on the graph of the equation, $(0,\text-3)$ is a solution to the equation. Any point not on the line is not a solution to the equation. 

Sometimes the coordinates of a solution cannot be determined exactly by looking at the graph. For example, when $x=1.5$, the $y$-value is somewhere between -2 and -3. If we have a value for one of the variables, we can use the equation to figure out the value of the other variable.

$\begin{aligned}2x-4y&=12 \\2(1.5)-4y&=12 \\3-4y&=12\\\text-4y&=9\\y&=\text-\frac94\text{ or } \text- 2\frac14\end{aligned}$

The equation can also be used to check whether a pair of values is a solution to the equation by seeing if the values make the equation true. For example, since the values $x=5$ and $y=2$ do not make the equation true, then the point $(5,2)$ is not a solution and does not lie on the line.