Graphing lines

Understanding What It Means to Graph a Line

In this chapter you already know what a linear function is and what slope and intercepts are. The goal now is to turn an equation of a line into its picture on the coordinate plane.

To “graph a line” means: draw all points $(x,y)$ in the plane that satisfy the line’s equation. Because a line is straight, once you know two correct points you can draw the entire line through them. Different forms of the equation make this easier or harder, so we will practice several methods.

We work in the Cartesian coordinate plane: a horizontal $x$‑axis, a vertical $y$‑axis, and points written as $(x,y)$.

General Strategy for Graphing Any Line

No matter what form the equation is in, you can always graph it using this basic template:

1. Rewrite the equation (if needed) into a convenient form.
2. Find at least two easy points on the line.
3. Plot those points accurately on the coordinate plane.
4. Use a ruler (or straightedge) to draw a line through them.
5. Extend the line across the grid and add arrowheads at both ends to show it goes on infinitely.

In practice, different forms suggest specific shortcuts. The rest of this chapter focuses on these practical shortcuts.

Graphing Lines from Slope–Intercept Form

Slope–intercept form is
$$
y = mx + b,
$$
where $m$ is the slope and $b$ is the $y$‑intercept.

You already know how to read $m$ and $b$; here we focus on turning them into a graph.

Step-by-step procedure

1. Plot the $y$‑intercept
• The point is $(0,b)$.
• On the $y$‑axis, move up if $b>0$ or down if $b<0$ and mark that point.
2. Use the slope to find another point
• Think of the slope $m$ as a fraction:
  $$
  m = \frac{\text{rise}}{\text{run}}.
  $$
  If $m$ is not a fraction, you can write it as one, for example = \tfrac{2}{1}$ or $-3 = \tfrac{-3}{1}$.
• From the $y$‑intercept, move:
• right by the “run” (positive direction on $x$),
• up if “rise” is positive, down if “rise” is negative.

Mark the new point.

3. Draw the line
• Use a straightedge to draw the line through the two points and extend in both directions.
• Put arrowheads at both ends.

Example

Graph $y = \dfrac{1}{2}x - 3$.

1. $b = -3$, so plot $(0,-3)$.
2. $m = \dfrac{1}{2}$, so rise $=1$, run $=2$.
• From $(0,-3)$, move 2 units right to $x=2$.
• Then move 1 unit up to $y=-2$.
• Plot $(2,-2)$.
3. Draw a straight line through $(0,-3)$ and $(2,-2)$.

Graphing Lines from Point–Slope Form

Point–slope form is
$$
y - y_1 = m(x - x_1),
$$
where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

You already know how to interpret this form; here is how to graph it directly.

Step-by-step procedure

1. Identify the given point and slope
• From the equation, read off $m$, $x_1$, and $y_1$.
• Be careful with signs: for example, $y + 3 = 2(x - 1)$ is $y - (-3) = 2(x - 1)$, so the point is $(1,-3)$.
2. Plot the given point $(x_1,y_1)$ on the coordinate plane.
3. Use the slope $m$ to find another point, just as with slope–intercept form:
• Rewrite $m$ as a fraction $\frac{\text{rise}}{\text{run}}$.
• From $(x_1,y_1)$, move according to rise and run.
4. Draw the line through the two points.

Example

Graph $y + 1 = -\dfrac{3}{2}(x - 4)$.

1. Rewrite: $y - (-1) = -\dfrac{3}{2}(x - 4)$, so
• Point: $(4,-1)$
• Slope: $m = -\dfrac{3}{2}$.
2. Plot $(4,-1)$.
3. Use $m = -\dfrac{3}{2}$:
• Rise $=-3$ means go 3 down.
• Run $=2$ means go 2 right.
• From $(4,-1)$, move 2 right to $x=6$, then 3 down to $y=-4$, giving point $(6,-4)$.
4. Draw the line through $(4,-1)$ and $(6,-4)$.

Graphing Lines from Standard Form

Standard form of a line is
$$
Ax + By = C,
$$
where $A$, $B$, and $C$ are constants and at least one of $A$, $B$ is nonzero.

You could solve for $y$ to convert to slope–intercept form, but there is a particularly simple method for graphing from standard form: use intercepts.

Intercept method

The intercepts of a line are where it crosses the axes:

• $x$‑intercept: point where the line crosses the $x$‑axis. There $y=0$.
• $y$‑intercept: point where the line crosses the $y$‑axis. There $x=0$.

For $Ax + By = C$:

• To find the $x$‑intercept, set $y=0$ and solve for $x$.
• To find the $y$‑intercept, set $x=0$ and solve for $y$.

These two intercepts are points on the line. Plot them and draw the line through them.

Step-by-step procedure

1. Find $x$‑intercept
• Set $y=0$ in $Ax + By = C$:
  $$
  Ax + B(0) = C \Rightarrow Ax = C \Rightarrow x = \frac{C}{A}\quad(\text{if }A \neq 0).
  $$
• The $x$‑intercept is $\left(\dfrac{C}{A}, 0\right)$.
2. Find $y$‑intercept
• Set $x=0$:
  $$
  A(0) + By = C \Rightarrow By = C \Rightarrow y = \frac{C}{B}\quad(\text{if }B \neq 0).
  $$
• The $y$‑intercept is $\left(0, \dfrac{C}{B}\right)$.
3. Plot both intercepts and draw the line through them.

If $A=0$ or $B=0$, one of the intercepts is especially easy (and leads to horizontal or vertical lines, discussed below).

Example

Graph $2x + 3y = 6$.

1. $x$‑intercept: set $y=0$:
   $$
   2x + 3(0) = 6 \Rightarrow x = 3.
   $$
   So the $x$‑intercept is $(3,0)$.
2. $y$‑intercept: set $x=0$:
   $$
   2(0) + 3y = 6 \Rightarrow y = 2.
   $$
   So the $y$‑intercept is $(0,2)$.
3. Plot $(3,0)$ and $(0,2)$, then draw the line through them.

Graphing Horizontal Lines

A horizontal line has equation
$$
y = k
$$
for some constant $k$. Every point on the line has the same $y$‑coordinate, and the slope is $0$.

How to graph $y = k$

1. On the $y$‑axis, locate the point $(0,k)$.
2. Through that point, draw a line that goes left and right, staying at the same height $y=k$.
3. Extend it across the grid and mark arrowheads on both sides.

Example

Graph $y = -2$.

• Plot $(0,-2)$.
• Draw a straight horizontal line through that point, extending left and right.

Graphing Vertical Lines

A vertical line has equation
$$
x = a
$$
for some constant $a$. Every point on the line has the same $x$‑coordinate. A vertical line does not represent $y$ as a function of $x$, but it is still a line in the plane.

How to graph $x = a$

1. On the $x$‑axis, locate the point $(a,0)$.
2. Through that point, draw a line that goes straight up and down, staying at the same $x=a$.
3. Extend it and mark arrowheads on both ends.

Example

Graph $x = 3$.

• Plot $(3,0)$.
• Draw a straight vertical line through that point, extending up and down.

Choosing Points When Graphing

Sometimes you may prefer to make a table of values instead of relying on slope or intercepts, especially if:

• the slope is messy (for example, involves fractions that are hard to count on your grid), or
• the equation has been rearranged and you are unsure of the form.

Table-of-values method (works for any line)

1. Choose at least two convenient $x$‑values (often small integers).
2. Plug each $x$ into the equation to find the corresponding $y$.
3. Record them as ordered pairs $(x,y)$.
4. Plot all your points and draw a straight line through them.

More than two points can be helpful to check for mistakes; they should all lie on the same straight line.

Example

Graph $y = 2x + 1$ using a table.

1. Choose $x = -1, 0, 1$.
2. Compute:
• If $x=-1$: $y = 2(-1) + 1 = -1$, so point $(-1,-1)$.
• If $x=0$: $y = 2(0) + 1 = 1$, so point $(0,1)$.
• If $x=1$: $y = 2(1) + 1 = 3$, so point $(1,3)$.
3. Plot the three points and draw the line through them.

Because a line is straight, you do not need many points, but two or three help confirm correctness.

Recognizing a Line from Its Graph

Graphing lines often goes together with reading information from a given graph.

Given a straight line drawn on a coordinate plane, you can:

• Find the $y$‑intercept: Look where the line crosses the $y$‑axis. If this is at $(0,b)$, then the $y$‑intercept is $b$.
• Find the $x$‑intercept: Look where the line crosses the $x$‑axis, at $(a,0)$.
• Estimate the slope:
• Pick two clear points on the line, say $(x_1,y_1)$ and $(x_2,y_2)$.
• Compute
  $$
  m = \frac{y_2 - y_1}{x_2 - x_1}.
  $$

Once you have the slope and intercept, you can often write the equation of the line in the form $y = mx + b$.

Common Mistakes and How to Avoid Them

1. Forgetting that slope is rise over run
• Move up/down (rise) and right/left (run) from a known point.
• Remember: “run” is usually taken to the right (positive $x$). If the slope is negative, the rise must be downward.
2. Only plotting one point
• A single point is not enough to determine the direction of the line.
• Always plot at least two points before drawing the line.
3. Not extending the line
• A line is infinite in both directions.
• Extend your line across the grid and add arrowheads.
4. Mixing up $x$ and $y$ for vertical and horizontal lines
• $y = k$ is horizontal.
• $x = a$ is vertical.
5. Misreading intercepts
• The $x$‑intercept is where the line crosses the $x$‑axis (so $y=0$).
• The $y$‑intercept is where it crosses the $y$‑axis (so $x=0$).

Summary of Graphing Techniques

Depending on the equation form, you can choose a convenient method:

• Slope–intercept form $y = mx + b$
• Plot $(0,b)$, then use slope $m$ to find a second point.
• Point–slope form $y - y_1 = m(x - x_1)$
• Plot $(x_1,y_1)$, then use slope $m$ to find another point.
• Standard form $Ax + By = C$
• Find intercepts by setting $y=0$ and $x=0$, plot both, then connect.
• Horizontal lines $y = k$
• Draw a horizontal line through $(0,k)$.
• Vertical lines $x = a$
• Draw a vertical line through $(a,0)$.
• Any form
• Make a small table of values, plot the points, and draw a straight line.

These tools let you move fluently between equations of lines and their graphs, a key skill used repeatedly in algebra and beyond.