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y-intercept

y-intercept

A point at which a graph intersects the y-axis.

 

 

See also

x-intercept, z-intercept

Key Formula

y-intercept=(0, b)where b=f(0)y\text{-intercept} = (0,\, b) \quad \text{where } b = f(0)
Where:
  • bb = The y-coordinate of the point where the graph crosses the y-axis
  • f(0)f(0) = The value of the function when x = 0

Worked Example

Problem: Find the y-intercept of the line y = 3x + 7.
Step 1: Set x equal to 0, since every point on the y-axis has x = 0.
x=0x = 0
Step 2: Substitute x = 0 into the equation.
y=3(0)+7y = 3(0) + 7
Step 3: Simplify to find the y-value.
y=0+7=7y = 0 + 7 = 7
Step 4: Write the y-intercept as a point.
(0, 7)(0,\, 7)
Answer: The y-intercept is the point (0, 7).

Another Example

This example uses a nonlinear (quadratic) equation rather than a straight line, showing that the same method — substituting x = 0 — works for any type of equation.

Problem: Find the y-intercept of the quadratic equation y = 2x² − 5x + 4.
Step 1: Set x equal to 0.
x=0x = 0
Step 2: Substitute x = 0 into the equation.
y=2(0)2−5(0)+4y = 2(0)^2 - 5(0) + 4
Step 3: Simplify each term.
y=0−0+4=4y = 0 - 0 + 4 = 4
Step 4: Write the y-intercept as a point.
(0, 4)(0,\, 4)
Answer: The y-intercept is (0, 4). For any polynomial written in standard form, the y-intercept is simply the constant term.

Frequently Asked Questions

What is the difference between the y-intercept and the x-intercept?
The y-intercept is where the graph crosses the y-axis, found by setting x = 0. The x-intercept is where the graph crosses the x-axis, found by setting y = 0. A line like y = 2x + 6 has a y-intercept at (0, 6) and an x-intercept at (−3, 0).
Is the y-intercept a point or just a number?
Strictly, the y-intercept is a point on the coordinate plane, written as (0, b). However, many textbooks and teachers refer to just the number b as 'the y-intercept' for convenience. Either usage is common, but on tests you should follow whatever format your course expects.
Can a graph have more than one y-intercept?
If the graph represents a function, it can have at most one y-intercept, because a function assigns only one output to each input (including x = 0). However, non-function relations — like circles or sideways parabolas — can cross the y-axis at two or more points, giving multiple y-intercepts.

y-intercept vs. x-intercept

y-interceptx-intercept
DefinitionPoint where the graph crosses the y-axisPoint where the graph crosses the x-axis
How to find itSet x = 0 and solve for ySet y = 0 and solve for x
Coordinate form(0, b)(a, 0)
Number possible for a functionAt most oneZero, one, or many
In slope-intercept form y = mx + bRead directly as bSolve −b/m

Why It Matters

The y-intercept appears throughout algebra, precalculus, and statistics. In slope-intercept form (y = mx + b), the value b is the y-intercept, making it one of the first things you identify when graphing a line or interpreting a linear model. In real-world contexts — such as a cost equation where x is the number of items produced — the y-intercept often represents the starting value or fixed cost when x = 0.

Common Mistakes

Mistake: Confusing the y-intercept with the slope in y = mx + b.
Correction: In y = mx + b, the slope is m (the coefficient of x) and the y-intercept is b (the constant). For example, in y = 4x + 3, the y-intercept is 3, not 4.
Mistake: Setting y = 0 instead of x = 0 when looking for the y-intercept.
Correction: Setting y = 0 finds the x-intercept. To find the y-intercept, set x = 0 and solve for y. Remember: the y-intercept sits on the y-axis, where x is always 0.

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