Slope-Intercept Form
In short: Slope-intercept form is the equation layout y = mx + b, in which the coefficient m shows how steeply a line leans and the constant b shows the height where it starts on the vertical axis.
8th GradeFunctions and Graphs8.EE.B.6
What is Slope-Intercept Form?
Lines written as \(y = mx + b\) are in slope-intercept form. Two numbers do all the work: \(m\) multiplies x and controls how sharply the line tilts, while \(b\) is tacked on at the end and fixes where the line meets the vertical axis.
Since both features sit in plain view, you can sketch the graph without a table: mark \((0, b)\), then step right by the run and up or down by the rise that \(m\) describes. Equations in other layouts can be solved for \(y\) to match this pattern.
How to spot it
- An equation has y by itself on the left and an x term plus a constant on the right.
- You are handed the lean of a line and a single point on it, and asked for the whole equation.
- A line must be sketched fast, or two lines compared for tilt.
- An equation such as 6x − 3y = 12 needs rearranging before its tilt and starting height can be read.
The method
- Gather the tilt \(m\) and one point of the line, ideally the one sitting on the vertical axis.
- If the point is elsewhere, get \(b\) from \(b = y – mx\) using that point.
- Write \(y = mx + b\) with your numbers inserted.
- To draw it, mark \((0, b)\), apply rise over run from \(m\) to reach a second point, and join them.
Slope-Intercept Form: solved example
Problem
A ramp profile leans at \(\frac{3}{4}\) and goes through the point \((8, 2)\). Give its equation in the \(y = mx + b\) layout.
Solution
Find \(b\): \(b = y – mx = 2 – \frac{3}{4}(8) = 2 – 6 = -4\).
Substitute: \(y = \frac{3}{4}x – 4\). Check with the point: \(\frac{3}{4}(8) – 4 = 2\).
Try it
Question 1
Rewrite \(6x – 3y = 12\) in slope-intercept form.
Show the answer
\(-3y = -6x + 12\), so \(y = 2x – 4\): slope 2, y-intercept \(-4\).
Question 2
Read off the tilt and the vertical-axis crossing of \(y = -\frac{x}{5} + 9\).
Show the answer
The slope is \(-\frac{1}{5}\) and the y-intercept is 9.
Slope-Intercept Form vs. Linear Equation
A linear equation is any equation whose graph is a straight line, written in many possible shapes. The slope-intercept version is one particular arrangement, solved for y, that exposes the slope and the starting value. Every such equation is linear, but not every linear equation looks that way until it is rearranged.
Keep going
Lessons
Worksheets and quizzes
- Graphing Linear Equations – Worksheet
- Graphing Linear Equations – Quiz
- Writing Linear Equations and Functions – Worksheet
- Writing Linear Equations and Functions – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Functions and Graphs