Explainer: Determining the Slope of a Line from a Graph, a Table, or Coordinates

In this explainer, we will learn how to find the slope of a line using graphs, tables, or coordinates.

For this, you need to already be familiar with functions.

A linear function is a special function whose equation is of the form 𝑦=π‘šπ‘₯+𝑏, where π‘š and 𝑏 are constants.

Let us fill in a function table for 𝑦=2π‘₯+3.

π‘₯012
𝑦357

We see that for π‘₯=0, the function value (i.e., the value of 𝑦) is 𝑏 (here 3). The point (0,𝑏) is the point where the line intersects with the 𝑦-axis, that is, the 𝑦-intercept. For this reason, the constant 𝑏 is often called the 𝑦-intercept (it is the 𝑦-coordinate of the 𝑦-intercept).

Then, when the value of π‘₯ increases by 1, the value of 𝑦 increases by 2, which comes from the number 2 in the 2π‘₯ in the equation. Hence, the way the value of 𝑦 increases when π‘₯ increases is given by the constant π‘š: when π‘₯ increases by 1, 𝑦 increases by π‘š.

This can be generalized with a function table for the general equation 𝑦=π‘šπ‘₯+𝑏.

Let us look now at the effect of π‘š on the graph of a linear function. The figure shows six lines whose equations are 𝑦=π‘šπ‘₯βˆ’1, with π‘š=1,2,3,βˆ’1,βˆ’2, or βˆ’3.

We observe that all lines intersect at the 𝑦-intercept of coordinates (0,βˆ’1). The absolute value of π‘š determines the steepness of the line: the higher the absolute value, the steeper the line, while its sign determines whether the line goes up (positive π‘š) or down (negative π‘š) when π‘₯ increases (i.e., when we go from the left to the right).

For this reason, π‘š is called the slope of the line: its absolute value determines how steep the slope is and its sign gives the direction of the slope.

As mentioned above, a linear function of the equation 𝑦=π‘šπ‘₯+𝑏 is characterized by the fact that whenever π‘₯ increases by 1, 𝑦 increases by π‘š. This means that if π‘₯ increases by 2, then 𝑦 increases by 2π‘š; if π‘₯ increases by 3, then 𝑦 increases by 3π‘š; and so on, as shown in the figure for two different lines.

In other words, the change in the 𝑦-coordinate between any two points is proportional to the change in the π‘₯-coordinate between the two points, and the coefficient of proportionality is π‘š. The slope π‘š of the line is therefore the rate of change of the function the line represents.

From this, it follows that π‘š is simply the rate of the vertical change (in the 𝑦-coordinate) to the horizontal change (in the π‘₯-coordinate) between any two distinct points: π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯, where 𝐴 and 𝐡 are two points lying on the line.

Let us summarize what we have just learned.

Characteristics of a Linear Function

The graph of a linear function is a straight line.

A linear function has a constant rate of change, which means that the difference in the 𝑦-coordinates of any two points on the straight line representing the linear function is proportional to the difference in their π‘₯-coordinates.

The rate of change is the slope of the line.

The equation of a line is generally written in the form 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is the 𝑦-intercept.

How to Find the Slope of a Line given Two Points

The slope of a line, π‘š, is the rate of the vertical change to the horizontal change between two points.

For two points 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦) lying on a line, the slope is then π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯.

Let us look at the first example in a real-world context.

Example 1: Finding the Slope of a Line in a Real-World Context

Find the slope of a road that rises 32 feet for every horizontal change of 400 feet.

Answer

It is said here that the road rises 32 feet for a horizontal change of 400 feet. This can be represented as shown in the figure.

The slope is the rate of the vertical change to the horizontal change, that is, 32400.

This fraction can be expressed in its simplest form by dividing both the numerator and the divisor by 16. The result is then 225. Hence, the slope of the road is 225.

In the next example, we are going to find the slope of a line from its graph.

Example 2: Finding the Slope of a Line from Its Graph

Find the slope of the line shown.

Answer

The line shown goes through the points of the coordinates (βˆ’5,1) and (βˆ’1,βˆ’4). The horizontal change from the first point to the second is βˆ’1βˆ’(βˆ’5)=4, and the vertical change is βˆ’4βˆ’1=βˆ’5.

The slope is the rate of the vertical change to the horizontal change, that is, βˆ’54.

Hence, the slope of the line is βˆ’54.

Now, we are going to find the slope a line from a function table.

Example 3: Finding the Slope of a Line from a Function Table

Find the slope of the line that describes the relationship between the perimeter (𝑦) and the length (π‘₯) as shown in the table.

Side Length (π‘₯)681012
Perimeter (𝑦)36486072

Answer

We are given here four points lying on a line representing the perimeter in terms of the side length of a certain shape. We know that the slope is the rate of the vertical change to the horizontal change between any two points. Using this, we can easily find the slope by choosing any two points, for example, (6,36) and (12,72). We find π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯=366=6.

We could have also noticed that there is a proportional relationship between π‘₯ and 𝑦: each 𝑦 value is indeed 6 times the π‘₯ value; that is, 𝑦=6π‘₯. The slope of the line representing this relationship is 6.

Example 4: Finding the Slope of a Line from a Function Table

Given that the points shown in the table lie on a line, determine the slope of the line.

π‘₯ 1 4 7 10
𝑦 15 19 23 27

Answer

We are given here four points lying on a line.

We know that the slope is the rate of the vertical change to the horizontal change between any two points. Using this, we can easily find the slope by choosing any two points, for example, (4,19) and (10,27). We find, for the slope π‘š, π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯=27βˆ’1910βˆ’4=86.

This fraction can be simplified to 43, which can be written as 113.

The slope of the line is 113.

Finally, we are going to find the slope of a line given two points on it.

Example 5: Finding the Slope of a Line given Two Points

Determine the slope of the line that passes through the points 𝐴(2,βˆ’5) and 𝐡(4,5).

Answer

We are given the coordinates of two points that lie on the line.

We know that the slope is the rate of the vertical change to the horizontal change between any two points. Hence, we find, for the slope π‘š, that π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯=5βˆ’(βˆ’5)4βˆ’2=102.

This fraction can be reduced to 5.

Hence, the slope of the line is 5.

Key Points

  1. The graph of a linear function is a straight line.
  2. A linear function has a constant rate of change, which means that the difference in the 𝑦-coordinates of any two points on the straight line representing the linear function is proportional to the difference in their π‘₯-coordinates.
  3. The rate of change is the slope of the line.
  4. The equation of a line is generally written in the form 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is the 𝑦-intercept.
  5. The slope of a line, π‘š, is the rate of the vertical change to the horizontal change between two points. For two points 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦) lying on a line, the slope is π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯.

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