Lines, Slope and Linear Equations

A line changes at a steady rate: the same step in $x$ always gives the same change in $y$. That makes lines the measuring stick for every other function in this course. To see how fast a curve changes between two of its points, you draw the line through them and read its slope.

Slope

The slope of a line is its rise over its run, the change in $y$ divided by the change in $x$, between any two of its points:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Subtract in the same order on top and bottom. For $(-2, 5)$ and $(4, -7)$: $m$$= \frac{-7 - 5}{4 - (-2)}$$= \frac{-12}{6}$$= -2$.

Equations of a line

Point-slope form uses one point $(x_1, y_1)$ and the slope $m$: $y - y_1 = m(x - x_1)$. It says that from the known point, $y$ changes by $m$ times the change in $x$. Solved for $y$, it is $y = y_1 + m(x - x_1)$, the form you will use to estimate values.

Slope-intercept form $y = mx + b$ has slope $m$ and $y$-intercept $(0, b)$.

When you rewrite point-slope form, distribute the slope to both terms: $y - 4 = -3(x + 2)$ becomes $y - 4 = -3x - 6$, so $y = -3x - 2$.

A line through two points of a curve

The two points do not have to come from a line. For $f(x) = x^2$, the points $(2, f(2)) = (2, 4)$ and $(5, f(5)) = (5, 25)$ are on a curve, and the line through them has slope $\frac{25 - 4}{5 - 2} = 7$. The next topic gives this slope its own name and uses it to measure how fast $f$ changes; the algebra is the same as here.

Slope in context

When $x$ and $y$ are quantities, the slope has units of output per input. For a phone plan $C(m) = 15 + 0.10m$ dollars for $m$ minutes, the slope is $0.10$ dollars per minute: each extra minute adds $0.10. The intercept, $15, is the cost at $m = 0$.