Find Equation of a Line from a Graph

Review of Formulas

The following formulas for slopes and equations of lines are used to solve the examples below:

The slope \(m\) of a line with rise \(\Delta y\) corresponding to run \(\Delta x\) is given by: \[ m = \dfrac{\Delta y}{\Delta x} \]
The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \dfrac{y_2 - y_1}{x_2 - x_1} \]
The equation of a line passing through point \((x_0, y_0)\) with slope \(m\) is given by: \[ y - y_0 = m(x - x_0) \]
The equation of a line with \(y\)-intercept at \((0, b)\) and slope \(m\) is given by: \[ y = mx + b \]
The equation of a horizontal line through \((x_0, y_0)\) is given by: \[ y = y_0 \]
The equation of a vertical line through \((x_0, y_0)\) is given by: \[ x = x_0 \]
The slopes \(m_1\) and \(m_2\) of two perpendicular lines satisfy: \[ m_1 \cdot m_2 = -1 \]
The slopes \(m_1\) and \(m_2\) of two parallel lines satisfy: \[ m_1 = m_2 \]

Examples with Detailed Solutions

Example 1: Graph of Line with Points

Find the equation of the line whose graph is shown below and write it in slope-intercept form.

graph of line for example 1 — Fig.1 - Line passing through graph points

View Solution to Example 1

Use two points \((2, 2)\) and \((3, 4)\) from the graph to find the slope \(m\):

\[ m = \dfrac{4 - 2}{3 - 2} = 2 \]

Using a point and the slope in point-slope form:

\[ y - 2 = 2(x - 2) \]

Simplifying to slope-intercept form:

\[ y = 2x - 2 \]

Example 2: Graph of Line Given Run and Rise

Find the equation of the line whose graph, including the rise and run, is shown below.

graph of line for example 2 — Fig.2 - Line with rise and run

View Solution to Example 2

The slope of the line is given by:

\[ m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{2}{2} = 1 \]

The \(y\)-intercept is the point \((0, -\frac{3}{2})\).

The equation in slope-intercept form is:

\[ y = x - \frac{3}{2} \]

Example 3: Graph of Line Given Run and Fall

Find the equation, in slope-intercept form, of the line whose graph, including the fall and run, is shown below.

graph of line for example 3 — Fig.3 - Line with fall and run

View Solution to Example 3

The slope of the line is given by:

\[ m = \dfrac{\text{fall}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{-4.5}{3} = -1.5 \]

Given the point \((1, -5)\) and slope, use point-slope form:

\[ y - (-5) = -1.5(x - 1) \]

In slope-intercept form:

\[ y = -1.5x - 3.5 \]

Example 4: Graph of Horizontal Line

Find the equation of the horizontal line whose graph is shown below.

graph of horizontal line for example 4 — Fig.4 - Horizontal line

View Solution to Example 4

The equation of the horizontal line passing through \((0, -3)\) is:

\[ y = -3 \]

Example 5: Graph of Vertical Line

Find the equation of the vertical line whose graph is shown below.

graph of vertical line for example 5 — Fig.5 - Vertical line

View Solution to Example 5

The equation of the vertical line passing through \((2, 0)\) is:

\[ x = 2 \]

Example 6: Graph of Perpendicular Line

Find the slope-intercept form of the equation of line \(L_2\) that is perpendicular to line \(L_1\) as shown below.

graph of perpendicular line for example 6 — Fig.6 - Perpendicular lines

View Solution to Example 6

First, find the slope \(m_1\) of line \(L_1\) passing through points like \((-4, 0)\) and \((2, 2)\):

\[ m_1 = \dfrac{2 - 0}{2 - (-4)} = \dfrac{2}{6} = \dfrac{1}{3} \]

Since \(L_1\) and \(L_2\) are perpendicular, their slopes satisfy \(m_1 \cdot m_2 = -1\):

\[ m_2 = -\dfrac{1}{m_1} = -\dfrac{1}{1/3} = -3 \]

Using point \((-4, 0)\) on \(L_2\) and slope \(m_2 = -3\):

\[ y - 0 = -3(x - (-4)) \implies y = -3x - 12 \]