Slope and Intercepts

From point-slope to slope-intercept form

Point-slope form asks for a point and a slope, and any point on the line will serve. Some points, though, are more convenient than others. The most convenient point of all is the one where the line crosses the $y$-axis, because its $x$-coordinate is $0$. That zero coordinate makes the algebra collapse to almost nothing. Building point-slope form on that point produces the cleanest equation of a line there is.

That crossing point has a name. The y-intercept is the point where the line crosses the $y$-axis. Every point on the $y$-axis has $x = 0$, so the $y$-intercept has the form $(0, b)$ for some number $b$. That number $b$ is the height at which the line meets the axis.

Reading the slope and the intercept

Once an equation is written as $y = mx + b$, the slope and the intercept require no work at all. In that form the slope $m$ is the number multiplying $x$, and $b$ is the constant, which fixes the $y$-intercept at $(0, b)$. The only care needed is to read each number with its own sign, and to make sure the equation is actually in this form before you read it.

Writing an equation is the same reading in reverse. If you are told a line’s slope and its $y$-intercept, you have $m$ and $b$ already, so drop them into $y = mx + b$ and you are done.

That reversal is what the figure below is for. Its three controls are $m$ and $b$ themselves, with the slope split into the rise and the run that make it. The figure’s readout writes the equation back out for whatever you set. So you can work the lesson in either direction: choose the numbers and read the equation, or pick an equation and hunt for the numbers that produce it.

Start at a rise of $3$ over a run of $4$ with the crossing at $-2$, and the readout says $y = (3/4)x - 2$. Now step the crossing up to $0$: the constant disappears from the equation, because a line through the origin has $b = 0$. The equation then reads $y = (3/4)x$ with nothing added. Take the rise down to $0$ next. The $x$ term vanishes instead and the readout reads $y = 0$, a flat line along the horizontal axis. That is the same shape as the table’s $y = 7$ row, a slope of $0$ with the constant left standing. But this figure’s crossing point stops at $6$, so the constant it can reach is not $7$ itself. Finally set the rise negative and watch the minus sign move into the coefficient rather than the constant. Those are the three places a sign or a missing term can hide.

Graphing from slope-intercept form

Slope-intercept form is built for fast graphing, because the two numbers it hands you are exactly the two things you need to draw a line. Those two things are a point to start from and a direction to head in. The recipe is short. Plot the $y$-intercept $(0, b)$ first, since it sits right on the $y$-axis and takes no calculation. Then read the slope as rise over run and step from that point to a second one. That step means moving across by the run and up or down by the rise. Two points fix a line, so lay a ruler across them and draw.

Converting to slope-intercept form

Not every equation arrives in slope-intercept form. Standard form $Ax + By = C$ and point-slope form $y - y_1 = m(x - x_1)$ both hide the slope and intercept until you rearrange them. Getting to $y = mx + b$ is always the same goal: solve the equation for $y$, so that $y$ stands alone on one side with a coefficient of $1$. Then the slope and intercept read straight off.

A point-slope equation converts the same way, by distributing the slope and then moving the constant across. Take $y - 3 = 2(x - 1)$. Distribute the $2$ on the right, then add $3$ to both sides:

The line has slope $2$ and $y$-intercept $(0, 1)$. Both forms describe one line; slope-intercept form just displays the slope and intercept that point-slope form kept folded up.

Finding the intercepts

The lesson is named for slope and intercepts, and slope-intercept form already hands you one of the two intercepts for free. The $y$-intercept is $(0, b)$, read straight from the constant, or found by setting $x = 0$ in any form of the equation. The other crossing, the x-intercept, is the point where the line meets the $x$-axis. Every point on the $x$-axis has $y = 0$, so to find it you set $y = 0$ and solve for $x$. The two rules are mirror images: zero the coordinate of the axis you are crossing, which is always the opposite letter from the one you keep.

Special cases

Three lines stretch slope-intercept form to its edges, or past them, and each is worth a moment.

A horizontal line has slope $0$. In slope-intercept form that reads $y = 0x + b$, which collapses to $y = b$. So a horizontal line is its own $y$-intercept turned into an equation: it meets the $y$-axis at $(0, b)$ and never tilts. When $b$ is not $0$, the line has no $x$-intercept, because it holds the height $b$ and never reaches the $x$-axis. When $b = 0$, the line is the $x$-axis itself and lies flat along it.

A vertical line is the one case slope-intercept form cannot describe. Its slope is undefined, so there is no $m$ to write, and its equation is $x = c$ instead. A vertical line has an $x$-intercept at $(c, 0)$ but no $y$-intercept, unless $c = 0$, when the line is the $y$-axis.

A line through the origin has $y$-intercept $(0, 0)$, so its $b$ is $0$ and its equation is simply $y = mx$. Here the $x$-intercept and the $y$-intercept are the same point, the origin, because setting $x = 0$ and setting $y = 0$ both land there. Because both intercepts land on that same point, plotting such a line from intercepts alone is not enough; step out to a second point using the slope.

Common mistakes