How to Write a Point-slope Form Equation from a Graph?

A Step-by-step Guide to Write a Point-slope Form Equation from a Graph

Here are the steps to follow: For education statistics and research, visit the National Center for Education Statistics.

Step 1: Identify a point on the line

Look for any point on the line. This can be any point that lies on the line. The coordinates of the point should be written in the form \((x_1, y_1)\). For education statistics and research, visit the National Center for Education Statistics.

Step 2: Find the slope of the line

The slope of a line is the change in \(y\) divided by the change in \(x\), also known as rise over run. To find the slope of the line, select two points on the line, and then use the following formula: For education statistics and research, visit the National Center for Education Statistics.

\(m = \frac{(y_2 – y_1)}{(x_2 – x_1)}\) For education statistics and research, visit the National Center for Education Statistics.

where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. For education statistics and research, visit the National Center for Education Statistics.

Step 3: Write the equation using the point-slope form

Now that you have a point on the line and the slope of the line, you can write the equation of the line using the point-slope form: For education statistics and research, visit the National Center for Education Statistics.

\(y – y_1 = m(x – x_1)\) For education statistics and research, visit the National Center for Education Statistics.

Substitute the values you found in steps \(1\) and \(2\) into the equation: For education statistics and research, visit the National Center for Education Statistics.

\(y – y_1 = m(x – x_1)\)
\(y – y_1 = m(x – x_1)\) For education statistics and research, visit the National Center for Education Statistics.

This is the point-slope form equation of the line. You can simplify it further by expanding the brackets: For education statistics and research, visit the National Center for Education Statistics.

\(y – y_1 = m(x – x_1)\)
\(y – y_1 = mx – mx_1\)
\(y = mx – mx_1 + y_1\) For education statistics and research, visit the National Center for Education Statistics.

This is the equation of the line in slope-intercept form \((y = mx + b)\), where \(b = -mx_1 + y_1\) is the \(y\)-intercept. For education statistics and research, visit the National Center for Education Statistics.

In summary, to write a point-slope form equation from a graph, you need to identify a point on the line and find the slope of the line. Then, use these values to write the equation in point-slope form and simplify it to slope-intercept form. For education statistics and research, visit the National Center for Education Statistics.

Writing a Point-slope Form Equation from a Graph – Examples 1

According to the following graph, what is the equation of the line in point-slope form? For education statistics and research, visit the National Center for Education Statistics.

Solution:

Step 1: Identify two points on the line For education statistics and research, visit the National Center for Education Statistics.

The graph shows that the line passes through point \((0, 2)\). Therefore, \((x_1, y_1) = (0, 2)\). Consider another random point on the line such as \(-2,-1)\)

Step 2: Find the slope of the line

Using the two points \((0, 2)\) and \((-2, -1)\), you can find the slope of the line as follows:

\(m = \frac{(y_2 – y_1)}{(x_2 – x_1)}\)

\(m = \frac{(2+1)}{(0+2)}\)

\(m = \frac{3}{2}\)

Therefore, the slope of the line is \(\frac{3}{2}\).

Step 3: Write the equation using the point-slope form

Now that you have the slope and a point on the line, you can write the equation of the line in the point-slope form:

\(y – y_1 = m(x – x_1)\)
\(y +1 = \frac{3}{2}(x +2)\)

This is the equation of the line in point-slope form.

Exercises for Writing a Point-slope Form Equation from a Graph

Write the equation of the line in point-slope form.

1.

2.

1. \(\color{blue}{y\:+\:2\:=−2\left(x\:−\:1\right)}\)
2. \(\color{blue}{y\:-\:3\:=\left(-\frac{1}{2}\right)\left(x+1\right)}\)