How to Find Slope From a Graph?

The slope of a line is defined as the change in the \(y\) coordinate relative to the change in the \(x\) coordinate of that line. In the following guide, you will learn about ways of calculating slope from a graph.

How to Find Slope From a Graph?

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of a line in the coordinate plane. In general, to find the slope of a line, we must have values of both different coordinates on the line.

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Step by step guide to finding slope from a graph

The process of finding the slope from a graph uses the slope formula \(\frac{rise}{run}\). When the graph of a line is given and we are asked to find its equation, the first thing that we need to do is to find its slope.

Finding slope from a graph

The slope of a line is the ratio of increase to execution. Hence, here are the steps to find the slope of the chart:

  • Step 1: Select any two random points on the graph of the line (preferably with integer coordinates).
  • Step 2: Label them as \(A\) and \(B\) (in any order).
  • Step 3: Calculate “rise” from \(A\) to \(B\). As we go from \(A\) to \(B\) vertically, if we have to go “up”, then the rise is positive; “down”, then the rise is negative.
  • Step 4: Now, use the formula: \(\color{blue}{slope =\frac{rise}{run}}\).

Here is an example of the graph of a line.

Here, we get \(A= (1, 1)\) and \(B= (0, 3)\). Note that here we have the points with integer coordinates. Make a right triangle that starts at \(A\) and ends at \(B\), which makes the process of finding rise and run easier. Here, we have to move vertically “up” to reach \(B\) from \(A\) and hence rise \(= +2\), and we have to move horizontally “left” to reach from \(A\) to \(B\) and hence run \(=-1\). So slope \(=\frac{ rise}{run} = \frac{2}{-1}= -2\).

We do not need to select these points just to calculate the slope, and we do not need to select them in the above order. Here you can see the graph of the same line where the same points are chosen in a different order and different points are chosen. Note that the line slope (final answer) will eventually be the same.

Calculating slope from a graph using the slope formula

The slope formula is used to find the slope of a line that joins two points \((x_1, y_1)\) and \((x_2, y_2)\). sing this formula, the slope of the line is, \(\color{blue}{m = \frac{(y_2 – y_1)}{ (x_2 – x_1)}}\). We can use the same formula to find the slope of a line from its graph also. For this:

  • Step 1: Select both points on the line from its graph.
  • Step 2: Represent them as \((x_1, y_1)\) and \((x_2, y_2)\) in any order.
  • Step 3: Apply the formula \(m = \frac{(y_2 – y_1)}{ (x_2 – x_1)}\) to find the slope.

Finding Slope from a Graph – Example 1:

Consider the above graphs and find the slope.

Solution:

Let’s choose two points \((-1, 5)\) and \((1, 1)\) on it. Now,

\((x_1, y_1) = (-1, 5)\)

\((x_2, y_2) = (1, 1)\)

Slope, \(m = \frac {(y_2 – y_1)}{(x_2 – x_1)}\)
\(=\frac{ (1-5)}{ (1-(-1))}\)
\(= -\frac{4}{2}\)
\(= -2\)

When we calculated the slope using the \(\frac{rise}{run}\), we had the same slope \((-2)\).

Exercises for Finding Slope From a Graph

Find the slope of the line from the following graphs using the \(\frac{rise}{run}\) formula.

This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{0}\)
  2. \(\color{blue}{-3}\)

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