Intercepts of Graphs

Q: Which variable do I set to zero for the $x$-intercept?

Set $y = 0$. The $x$-intercept sits on the x-axis, where the height is zero. It feels backwards, so check with the picture: at a point like $(3, 0)$, it is the $y$-coordinate that vanished.

Q: Is the $y$-intercept the number b in y = mx + b?

Yes. Substituting $x = 0$ into $y = mx + b$ leaves $y = b$, so the constant term is the $y$-intercept. That shortcut only works once the equation is solved for $y$ — in a form like $4x - 3y = 12$ you still need to substitute.

Intercepts are the points where a graph crosses the axes. The $x$ -intercept is where the graph crosses the x-axis, and at that moment the height is zero — so $y = 0$ there. The $y$ -intercept is where the graph crosses the y-axis, and since that axis sits at horizontal position zero, $x = 0$ there.

That single idea — one variable is zero at each intercept — is the whole technique. On a graph you just look for the crossing points. From an equation, you set one variable to zero and solve for the other.

Reading intercepts from a graph

Follow the graph until it touches an axis. Where it crosses the x-axis, read the $x$ -value: the line below crosses at $(3, 0)$ , so its $x$ -intercept is $3$ . Where it crosses the y-axis, read the $y$ -value: this line crosses at $(0, -2)$ , so its $y$ -intercept is $-2$ .

Written as points, an $x$ -intercept always looks like $(a, 0)$ and a $y$ -intercept always looks like $(0, b)$ . If a point has neither coordinate equal to zero, it is not an intercept.

Finding intercepts from an equation

To find the $x$ -intercept, set $y = 0$ and solve for $x$ . To find the $y$ -intercept, set $x = 0$ and solve for $y$ . The rule is: zero out the other variable — the one you are not solving for.

This is where most mistakes happen, because it feels backwards. The $x$ -intercept comes from setting $y = 0$ , not $x = 0$ . Picture the graph: on the x-axis, it is the height $y$ that equals zero.

What intercepts mean in context

In a word problem, the $y$ -intercept is the starting value — the output when the input is zero. The $x$ -intercept is where the output runs out — the input that makes the quantity hit zero. If $y = 150 - 25x$ gives the money left on a gift card after $x$ weeks, the $y$ -intercept $150$ is the starting balance, and the $x$ -intercept $6$ is the week the card hits zero.

Worked examples

Example 1: both intercepts from standard form

Find the intercepts of $5x + 2y = 10$ .

For the

x

-intercept, set

y = 0

5x + 2(0) = 10

Solve

5x = 10, \quad x = 2

For the

y

-intercept, set

x = 0

5(0) + 2y = 10

Solve

2y = 10, \quad y = 5

Answer: $x$ -intercept $(2, 0)$ , $y$ -intercept $(0, 5)$

Example 2: intercepts from slope-intercept form

Find the intercepts of $y = 2x - 6$ .

The

y

-intercept is the constant: when

x = 0

y = 2(0) - 6 = -6

For the

x

-intercept, set

y = 0

0 = 2x - 6

Add

6

to both sides

6 = 2x

Divide by

2

x = 3

Answer: $x$ -intercept $(3, 0)$ , $y$ -intercept $(0, -6)$

Example 3: intercepts in context

A gift card balance is $y = 150 - 25x$ dollars after $x$ weeks. What do the intercepts mean?

y

-intercept: set

x = 0

y = 150 - 25(0) = 150

The card starts with $

150

x

-intercept: set

y = 0

0 = 150 - 25x

Solve

25x = 150, \quad x = 6

The card runs out after

6

weeks

Answer: Starting balance $ $150$ ; the balance reaches zero at week $6$

Try one yourself

Common questions

Which variable do I set to zero for the

x

-intercept?

Set $y = 0$ . The $x$ -intercept sits on the x-axis, where the height is zero. It feels backwards, so check with the picture: at a point like $(3, 0)$ , it is the $y$ -coordinate that vanished.

Can a graph have more than one

x

-intercept?

Yes. A line has at most one, but a parabola can cross the x-axis twice, and other curves can cross even more. Every crossing of the x-axis counts as an $x$ -intercept. A function can only have one $y$ -intercept, though — one input of $x = 0$ gives one output.

Is the

y

-intercept the number b in y = mx + b?

Yes. Substituting $x = 0$ into $y = mx + b$ leaves $y = b$ , so the constant term is the $y$ -intercept. That shortcut only works once the equation is solved for $y$ — in a form like $4x - 3y = 12$ you still need to substitute.

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Intercepts of Graphs

Reading intercepts from a graph

Finding intercepts from an equation

What intercepts mean in context

Worked examples

Example 1: both intercepts from standard form

Example 2: intercepts from slope-intercept form

Example 3: intercepts in context

Try one yourself

Common questions

Keep learning

Graphing Functions

Shapes of Graphs

Function Notation

Sketching and Comparing Functions