Slope-Intercept Form: y = mx + b
Graph lines using slope and y-intercept. Identify m and b from an equation in slope-intercept form and interpret them in context. Write equations in slope-intercept form given a graph or two points.
A lesson of Algebra 1, in Linear Functions, about 30 minutes.
Slope-Intercept Form
By the end of this lesson, you will be able to read the slope and intercept straight off $y = mx + b$, graph a line from it in two moves, rearrange an equation into it, and say what each letter means in a real situation.
Two Numbers Describe a Whole Line
$$y = mx + b$$
$$m = \text{the slope} \qquad b = \text{the } y\text{-intercept}$$
A straight line goes on forever in both directions, and those two numbers pin all of it. $b$ says where it crosses the vertical axis; $m$ says what it does from there.
$$y = 2x + 3 \qquad \text{starts at } 3, \text{ climbs } 2 \text{ per step}$$
Why the intercept is where $x = 0$. Put $x = 0$ into the rule and the $mx$ term vanishes, leaving $y = b$. So $b$ is not a separate fact to memorize, it is the output at zero, and in a real situation it is almost always the thing you pay or start with before anything else happens.
Graphing It in Two Moves
- Put a dot at $b$ on the vertical axis
- Use $m$ as a staircase from there: rise over run, and repeat
For $y = 2x + 3$: dot at $3$, then right $1$ up $2$, again, again. Join them.
For a fractional slope the staircase is the easy way. $y = \tfrac23 x - 1$ means dot at $-1$, then right $3$ up $2$. Using thirds of a square would be guesswork; the fraction is telling you exactly which whole squares to count.
Negative slope: right and DOWN. $y = -\tfrac12 x + 4$ is dot at $4$, then right $2$ down $1$.
Example 1: Reading and Graphing
Here is the whole method on one equation, worked all the way through. $$y = 3x - 2$$
- What is the slope? 3
- What is the y-intercept? -2
- The first dot goes at which point? Write it as a coordinate pair (0, -2)
- From there, going right 1, how far up? 3
Plot It: Two Points Make the Line
Worked Example 1: A Fractional Slope
$$y = \frac{2}{3}x + 1$$
Getting It Into That Form
An equation is often handed to you in some other arrangement. Solve for $y$ and the two numbers appear.
$$2x + y = 7 \longrightarrow y = -2x + 7$$
$$4x + 2y = 10 \longrightarrow 2y = -4x + 10 \longrightarrow y = -2x + 5$$
The step people drop is dividing EVERY term. Going from $2y = -4x + 10$ to $y = -4x + 5$ halves the constant and forgets the $x$ term, and the line that comes out is not the line you were given.
$$y - 3 = 5(x + 2) \longrightarrow y - 3 = 5x + 10 \longrightarrow y = 5x + 13$$
Multiply out, then isolate. The same two moves every time.
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