4.5 Use the Slope-Intercept Form of an Equation of a Line

Understanding the Slope-Intercept Form

Slope-Intercept Form and Graphs

The slope-intercept form of a linear equation is $y = mx + b$. This single equation tells you two things about a line right away:

• $m$ is the slope, which describes the steepness and direction of the line. A positive slope means the line rises from left to right. A negative slope means it falls from left to right.
• $b$ is the y-intercept, the point $(0, b)$ where the line crosses the y-axis.

Identifying Slope and Y-Intercept

To pull the slope and y-intercept out of an equation, make sure it's in $y = mx + b$ form. The coefficient of $x$ is the slope, and the constant term is the y-intercept.

For example, in $y = -3x + 2$:

• The slope is $-3$ (the line falls from left to right)
• The y-intercept is $2$, so the line crosses the y-axis at $(0, 2)$

The slope also represents the rate of change: for every 1 unit you move to the right along the x-axis, the y-value changes by $m$ units.

Graphing Lines with Slope-Intercept

To graph a line from slope-intercept form:

1. Plot the y-intercept $(0, b)$ on the y-axis.
2. Use the slope to find a second point. Write the slope as a fraction $\frac{\text{rise}}{\text{run}}$. From the y-intercept, move up (or down if negative) by the rise and right by the run. For example, a slope of $-3$ can be written as $\frac{-3}{1}$, so you'd move down 3 units and right 1 unit.
3. Draw a straight line through the two points, extending it in both directions.

Plotting at least two points is all you need, but finding a third point is a good way to check your work.

Efficient Line Graphing Methods

• When the y-intercept is a whole number, the standard method above works smoothly.
• When the y-intercept is a fraction or decimal, it can be easier to also find the x-intercept. Set $y = 0$ and solve for $x$. Then plot both intercepts and draw the line through them.
• Vertical lines can't be written in slope-intercept form because their slope is undefined. Their equation is $x = a$, where $a$ is a constant. Just draw a vertical line through the point $(a, 0)$.

Real-World Applications of Slope-Intercept

In real-world problems, the slope and y-intercept usually have concrete meanings.

Consider a cost equation like $C = 50x + 1000$, where $C$ is total cost and $x$ is the number of items produced:

• The slope $50$ is the cost per item (each additional item adds $50).
• The y-intercept $1000$ is the fixed cost, the amount you pay before producing anything (like equipment or setup fees).

Whenever you see a linear equation in context, ask yourself: What does the slope mean per unit? What does the y-intercept represent at the start?

Parallel Lines and Slopes

Parallel lines have the same slope but different y-intercepts. Because their steepness is identical, they never intersect.

For example, $y = 2x + 3$ and $y = 2x - 1$ are parallel. Both have a slope of $2$, but they cross the y-axis at different points ($(0, 3)$ and $(0, -1)$).

Perpendicular Lines and Slopes

Perpendicular lines meet at a 90° angle. Their slopes are negative reciprocals of each other. That means you flip the fraction and change the sign.

• If one line has a slope of $\frac{2}{3}$, a perpendicular line has a slope of $-\frac{3}{2}$.
• If one line has a slope of $-4$ (which is $\frac{-4}{1}$), a perpendicular line has a slope of $\frac{1}{4}$.

A quick check: multiply the two slopes together. If the product is $-1$, the lines are perpendicular. For the first example: $\frac{2}{3} \times -\frac{3}{2} = -1$.