How to Find Y-Intercepts

Last reviewed October 15, 2026

The y-intercept is the point where a graph meets the y-axis. Every point on that axis has x = 0, so the method is always the same: put x = 0 into the equation and work out y. It is usually the easiest point on a graph to find — and the one that most often tells you what the numbers mean.

The rule

Set x = 0 and solve for y. If the answer is b, the y-intercept is the point (0, b). For a function y = f(x), it is simply f(0).

For y = x² − 5x + 6, putting x = 0 leaves y = 6, so the y-intercept is (0, 6). Notice that every term with an x in it vanished: only the constant survives. That is the shortcut for any polynomial, and it is why the y-intercept is often called the initial value or starting value.

Lines in any form

Slope-intercept form

In y = mx + b, b is the y-intercept, which is where the form gets its name. y = 3x − 4 crosses the y-axis at (0, −4); y = −½x + 7 at (0, 7). Read the sign along with the number: y = 3x − 4 has b = −4, not 4.

Standard form

For Ax + By = C, put x = 0: By = C, so the y-intercept is (0, C/B). For 3x − 4y = 12, that is −4y = 12, giving (0, −3). Doing the same with y = 0 gives the x-intercept, (4, 0), and the two points together are the fastest way to draw the line.

Point-slope form

y − 5 = 2(x − 3) hides b. Put x = 0: y − 5 = −6, so y = −1 and the intercept is (0, −1). Expanding to y = 2x − 1 shows the same thing.

Horizontal and vertical lines

The horizontal line y = 4 crosses the y-axis at (0, 4). The vertical line x = 3 runs parallel to the y-axis and never meets it, so it has no y-intercept; x = 0 is the y-axis itself.

From two points

Given two points on a line, find the slope, then use either point to find b. Through (2, 5) and (6, 13):

1. Slope: m = (13 − 5)/(6 − 2) = 8/4 = 2.
2. Put (2, 5) into y = 2x + b: 5 = 4 + b, so b = 1.
3. The y-intercept is (0, 1). Check with the other point: 2·6 + 1 = 13.

A shortcut: b = y₁ − m·x₁. You are walking back along the line from the known point to x = 0, x₁ steps at a time, each step changing y by m.

From a table or a graph

A table that includes x = 0

Read it off: the y-value in the x = 0 row is the y-intercept.

A table that does not

If the x-values go up in equal steps and the y-values change by the same amount each time, the data is linear: work backwards to x = 0. In the table below, each step of 1 in x adds 3 to y, so going from x = 1 back to x = 0 subtracts 3 from 7.

The y-intercept is (0, 4), and the line is y = 3x + 4. If the y-values are multiplied by the same number each step instead, the data is exponential, and you divide backwards rather than subtract. For real data that is only roughly linear, fit a line first; the linear regression guide shows how, and the intercept of the fitted line is the estimate.

A graph

Find where the curve crosses the vertical axis and read the value. A graph gives a good estimate but rarely an exact one, so if you have the equation, use it to confirm.

Polynomials and exponentials

• Any polynomial: the constant term. 2x³ − x + 7 has its y-intercept at (0, 7). If there is no constant term, as in x³ − 2x, the graph passes through the origin.
• Factored form: put x = 0 into each bracket and multiply. (x − 2)(x + 3) gives (−2)(3) = −6, so (0, −6). No need to expand first.
• Vertex form: y = (x − 3)² + 1 gives 9 + 1 = 10. The vertex is (3, 1), but the y-intercept is (0, 10); they are different points unless the vertex sits on the y-axis.
• Exponentials: y = a·bˣ has y-intercept (0, a), because b⁰ = 1 for any base. y = 3·2ˣ starts at 3. With a shift, y = 2ˣ + 5, it is 1 + 5 = 6.
• Trig: y = sin x passes through the origin; y = cos x through (0, 1); y = 3 sin x + 2 through (0, 2).

Why a function has at most one

A function gives exactly one output for each input. The y-axis is the set of points with x = 0, so a function can meet it at most once, at (0, f(0)) — this is the vertical line test applied to the y-axis itself.

Equations that are not functions are not bound by this. The circle x² + y² = 25 gives y² = 25 when x = 0, so it meets the y-axis twice, at (0, 5) and (0, −5). A sideways parabola such as x = y² − 4 meets it at (0, 2) and (0, −2).

When there is no y-intercept

A graph has no y-intercept exactly when it has no point with x = 0. That happens in four ways:

• A vertical asymptote on the y-axis. y = 1/x and y = 1/x² shoot off to infinity as x approaches 0. See how to find vertical asymptotes.
• A domain that stops short of 0. y = ln x needs x > 0, and y = √(x − 2) needs x ≥ 2.
• A hole at x = 0. (x² + x)/x cancels to x + 1, which would give 1 at x = 0 — but the original fraction is 0/0 there, so the point (0, 1) is missing from the graph. The function has an x-intercept at (−1, 0) and no y-intercept.
• A vertical line other than the y-axis, such as x = 3.

What the y-intercept means

When x measures time, x = 0 is the start, and the y-intercept is the value at the start. That makes it the number in a word problem most often asked about:

• A taxi charges C = 2.5d + 4 for d miles: the y-intercept 4 is the flat fee you pay before moving.
• A population modelled by P = 1200·1.05ᵗ started at 1200.
• A ball thrown from a ledge, h = −16t² + 40t + 24, was released at a height of 24 feet.

Sometimes the intercept is meaningless. A regression of weight against height may give a negative y-intercept: the weight of someone 0 inches tall. The line fits the data where the data is and says nothing about x = 0.

Mistakes that cost marks

1. Setting y = 0 instead of x = 0. That finds the x-intercepts; see how to find x-intercepts.
2. Dropping the sign of b. y = 2x − 5 crosses at (0, −5).
3. Reading b from a line not in y = mx + b form. In 2x + 3y = 6, the 6 is not the y-intercept; (0, 2) is.
4. Writing a number when a point is asked for: (0, 4), not 4.
5. Simplifying first and reporting a y-intercept where the original function has a hole.
6. Confusing the vertex with the y-intercept of a parabola.

Questions

Find the intercepts of any function or equation