Drawing Lines from Slope and Intercepts, foundations for calculus lesson

A ramp that climbs 1 meter for every 12 meters of floor climbs at the same steepness everywhere along it. Measure a short stretch or a long one and the ratio of climb to distance is the same. That fixed ratio is the slope, and it is the reason the ramp is straight. Calculus is built on lines like this one, because when you zoom in near a single point, a smooth curve looks almost exactly like a line.

Key terms

Straight line

Definition A straight line in the coordinate plane is the set of all points $(x, y)$ that satisfy an equation

Ax + By = C,

where $A$ , $B$ and $C$ are real numbers and $A$ and $B$ are not both zero. When $B \ne 0$ the line is nonvertical, and the ratio of the change in $y$ to the change in $x$ between any two distinct points of it is the same number. When $B = 0$ the number $A$ is not zero, and the equation becomes the vertical line $x = \frac{C}{A}$ , where every point has the same $x$ value.

In plain words A line is a graph that never bends. Unless it is vertical, it goes up or down by the same amount for each step to the right, wherever you measure.

Example The points $(0, 1)$ , $(1, 3)$ , $(2, 5)$ and $(4, 9)$ all satisfy $y = 2x + 1$ . From $(0, 1)$ to $(1, 3)$ the graph climbs $2$ while moving $1$ to the right. From $(1, 3)$ to $(4, 9)$ it climbs $6$ while moving $3$ to the right. Both ratios equal $2$ , so the small triangle and the large triangle drawn under the line have the same shape.

Rise and run

Definition Moving from a point $P_1 = (x_1, y_1)$ to a point $P_2 = (x_2, y_2)$ , the run is $x_2 - x_1$ and the rise is $y_2 - y_1$ . Both are signed numbers. A negative run means a move to the left, and a negative rise means a move down.

In plain words The run is how far you go sideways and the rise is how far you go up. Going left or going down gives a negative number.

Example From $(3, 7)$ to $(-1, -1)$ the run is $-1 - 3 = -4$ and the rise is $-1 - 7 = -8$ . You move $4$ units left and $8$ units down.

Slope

Definition The slope of the line through two points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \ne x_2$ is

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

The numerator and the denominator must use the same order of the points. Reversing both gives the same number, but reversing only one changes the sign. For a nonvertical line the slope does not depend on which two distinct points you choose. If $x_1 = x_2$ the formula would divide by zero and is not defined.

In plain words Slope is rise divided by run. It tells you how much $y$ changes for each step of $1$ in $x$ . A positive slope means the line goes up to the right, and a negative slope means it goes down to the right.

Example Take the points $(-2, 5)$ and $(4, -1)$ . Subtracting in the same order gives

m = \frac{-1 - 5}{4 - (-2)} = \frac{-6}{6} = -1.

A common slip subtracts the $y$ values in one order and the $x$ values in the other, which gives $\frac{-6}{-6} = 1$ and the wrong sign.

Slope-intercept form

Definition Every nonvertical line can be written in exactly one way as

y = mx + b,

where $m$ is its slope and $b$ is the $y$ value at $x = 0$ . Conversely, for any real numbers $m$ and $b$ the graph of $y = mx + b$ is a nonvertical line with slope $m$ that passes through $(0, b)$ . A vertical line has no equation of this form.

In plain words Start at height $b$ on the $y$ axis. Then every step of $1$ to the right raises the line by $m$ .

Example To draw $y = \frac{2}{3}x - 1$ , start at $(0, -1)$ . The slope $\frac{2}{3}$ says run $3$ and rise $2$ , which lands on $(3, 1)$ . Check it in the rule, since $\frac{2}{3} \cdot 3 - 1 = 1$ . Draw the line through these two points.

To build a line from two points instead, find the slope and then $b$ . For $(-2, 5)$ and $(4, -1)$ the slope is $-1$ , so $y = -x + b$ . Putting in $(-2, 5)$ gives $5 = 2 + b$ , so $b = 3$ and the line is $y = -x + 3$ . The other point checks, since $-4 + 3 = -1$ .

Intercepts

Definition An intercept is a point where a graph meets a coordinate axis. A $y$ -intercept is a point of the graph of the form $(0, y)$ , and an $x$ -intercept is a point of the form $(x, 0)$ . For the line $y = mx + b$ the $y$ -intercept is $(0, b)$ . If $m \ne 0$ its only $x$ -intercept is

\left(-\frac{b}{m}, 0\right).

The coordinates of an intercept can be negative or zero. People often call the number $b$ itself the intercept.

In plain words The intercepts are where the line crosses the axes. Set $x = 0$ to find where it crosses the $y$ axis, and set $y = 0$ to find where it crosses the $x$ axis.

Example For $y = -3x - 6$ the $y$ -intercept is $(0, -6)$ . Setting $y = 0$ gives $0 = -3x - 6$ , so $x = -2$ and the $x$ -intercept is $(-2, 0)$ . Both intercepts lie on the negative parts of the axes, which is perfectly fine.

Horizontal line

Definition A horizontal line is the graph of an equation $y = c$ for a real constant $c$ . Its slope is $0$ , because any two of its points have the same $y$ value. If $c \ne 0$ it has no $x$ -intercept, and the line $y = 0$ is the $x$ axis itself.

In plain words The height never changes. Every point on the line has the same $y$ coordinate.

Example The points $(-3, 4)$ and $(5, 4)$ give

m = \frac{4 - 4}{5 - (-3)} = \frac{0}{8} = 0,

so the line through them is $y = 4$ . Its $y$ -intercept is $(0, 4)$ and it never meets the $x$ axis.

Vertical line

Definition A vertical line is the graph of an equation $x = c$ for a real constant $c$ . Any two of its distinct points have the same $x$ value, so the run is $0$ and the slope is undefined. A vertical line is not the graph of a function of $x$ , since the single input $c$ is paired with every $y$ .

In plain words The line goes straight up and down. Rise over run would mean dividing by zero, so it has no slope at all. That is different from slope zero, which belongs to a horizontal line.

Example The points $(2, -1)$ and $(2, 6)$ have run $2 - 2 = 0$ , so the slope formula does not apply. The line through them is $x = 2$ . Its $x$ -intercept is $(2, 0)$ and it has no $y$ -intercept.

Check yourself

Find the line through $(1, -4)$ and $(-3, 4)$ in slope-intercept form, and give both intercepts.

Show the answer

The slope is

m = \frac{4 - (-4)}{-3 - 1} = \frac{8}{-4} = -2.

So $y = -2x + b$ , and the point $(1, -4)$ gives $-4 = -2 + b$ , so $b = -2$ . The line is $y = -2x - 2$ . The $y$ -intercept is $(0, -2)$ , and setting $y = 0$ gives the $x$ -intercept $(-1, 0)$ .

Starting at the $y$ -intercept of $y = \frac{3}{4}x + 2$ , use the slope to find a second point, then find the $x$ -intercept.

Show the answer

Start at $(0, 2)$ . Run $4$ and rise $3$ to reach $(4, 5)$ , and indeed $\frac{3}{4} \cdot 4 + 2 = 5$ . Setting $y = 0$ gives $\frac{3}{4}x = -2$ , so the $x$ -intercept is $\left(-\frac{8}{3}, 0\right)$ .

Find an equation of the line through $(5, 1)$ and $(5, -3)$ . What is its slope?

Show the answer

Both points have $x = 5$ , so the line is the vertical line $x = 5$ . The run is $0$ , so its slope is undefined, and it cannot be written as $y = mx + b$ .

Drawing Lines from Slope and Intercepts

Key terms

Straight line

Rise and run

Slope

Slope-intercept form

Intercepts

Horizontal line

Vertical line

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus