Slope Intercept Form

Slope-intercept form means writing a non-vertical line as $y = mx + b$. In this equation, $m$ is the slope and $b$ is the y-intercept, so you can read the line's steepness and starting point right away.

That gives you a quick graphing rule: plot $(0, b)$ first, then use the slope to find another point. If $m > 0$, the line rises from left to right. If $m < 0$, it falls. If $m = 0$, the line is horizontal.

Use The Explorer To Separate $m$ And $b$

Move one slider at a time: change $m$ to see the line pivot, then change $b$ to see the whole line slide up or down without changing tilt.

How To Graph Slope-Intercept Form

For a line in slope-intercept form, graphing usually takes two steps:

1. Plot $(0, b)$ on the $y$-axis.
2. Use the slope as rise over run to locate another point.

For example, if $m = 3$, move right $1$ and up $3$. If $m = -2$, move right $1$ and down $2$. This works for non-vertical lines. A vertical line has equation $x = c$, so it cannot be rewritten as $y = mx + b$.

Worked Example: $y = 2x - 3$

Here $m = 2$ and $b = -3$.

The y-intercept is $(0, -3)$, so start there. Since the slope is $2$, you can read it as $2/1$: move right $1$ and up $2$. That gives another point at $(1, -1)$. Repeating the same move gives $(2, 1)$.

Those points all lie on the same line, so the graph rises steadily as $x$ increases. You can check them in the equation:

Set $m = 2$ and $b = -3$ in the explorer, then compare the intercept and sample points to the graph. That is the quickest way to connect the equation to the picture.

What To Notice In The Graph

1. Changing only $m$ changes the steepness and direction.
2. Changing only $b$ shifts the line up or down.
3. Lines with the same slope are parallel.
4. A larger absolute value of $m$ means a steeper line.

Try A Similar Line

Try your own version by choosing a line such as $y = -\frac{1}{2}x + 4$. Predict the intercept and whether the line rises or falls before you touch the widget, then use the graph to check your prediction.