Examples on how to find the equations of lines given their graphs. Examples and exercises are presented along with their detailed solutions and answers.

These following are reviews of formulas for slopes and equations of lines and are therefore used to solve the examples below.

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

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.

__Solution to Example 1__

Let use two points \( (2,2) \) and \( (3,4) \) from the graph to find the slope \( m \) of the line whose graph is shown above

\( m = \dfrac{4-2}{3-2} = 2\)

We know at least a point and the slope, the equation of the line in point slope form is given by

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

In point slope form the equation is written as

\( y = 2x - 2 \)

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.

__Solution to Example 2__

The slope of the line whose graph is shown above is given by

\( m = \dfrac{rise}{run} = \dfrac{\Delta y}{\Delta x} = \dfrac{2}{2} = 1\)

The y intercept is the point \( (0, - 3/2) \)

The equation of the line in slope intercept form is given by

\( y = x - 3/2 \)

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.

__Solution to Example 3__

The slope of the line whose graph is shown above is given by

\( m = \dfrac{fall}{run} = \dfrac{\Delta y}{\Delta x} = \dfrac{-4.5}{3} = -1.5\)

Given a point \( (1,-5) \) and the slope of the line, we use the point slope form of the equation of the line

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

In slope intercept form, the equation of the above graph is written as

\( y = -1.5 x - 3.5 \)

Example 4 Graph of horizontal line

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

__Solution to Example 4__

The equation of the horizontal line through the point \( (0,-3) \) is given by

\( y = - 3 \)

Example 5 Graph of vertical line

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

__Solution to Example 5__

The equation of the vertical line through the point \( (2,0) \) is given by

\( x = 2 \)

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.

__Solution to Example 6__

We need to find the slope \( m_2 \) of line \( L_2 \) which is perpendicular to line \( L_1 \) with slope \( m_1 \) given by

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

\( L_1 \) and \( L_2 \) are perpendicular and therefore

\( m_1 \cdot m_2 = -1 \)

Solve the above for \( m_2 \)

\( m_2 = - 1 / m_1 = - 1 / (1/3) = - 3\)

The equation of line \( L_1 \) in point slope form is given by

\( y - 0 = - 3(x - (-4)) \)

and in slope intercept form is given by

\( y = - 3 x - 12 \)

Find the equations of the lines \(L_1\), \( L_2 \), \( L_3 \) and \( L_4 \) such that \(L_2 \) is parallel to \( L_1 \), \(L_3 \) is perpendicular to \(L_1\) and \( L_4 \) is a horizontal line.

__Answers to the Above Exercises__

- Equation of \(L_1\) : \( y = 2 x + 4 \)
- Equation of \(L_2\) : \( y = 2x - 2.5 \)
- Equation of \(L_3\) : \( y = - \dfrac{1}{2} x - \dfrac{7}{2} \)
- Equation of \(L_4\) : \( y = - 5 \)

Equation of Line Questions with Solutions.

Slopes of Parallel Lines Questions.

Slopes of Perpendicular Lines Questions.

slopes