What Is the Y Intercept?
The y intercept of a graph is the point where the graph crosses, or touches, the y-axis. Because every point on the y-axis has an $x$-coordinate of 0, the y intercept always has the form $(0, c)$ for some number $c$. In everyday use, people often call the number $c$ alone "the y intercept", with the understanding that the actual point is $(0, c)$.
A function has at most one y intercept, because plugging in $x = 0$ gives a single $y$-value (or none, if the function is undefined there). A relation that is not a function — a circle, say — can cross the y-axis twice.
How to Find the Y Intercept: the Universal Method
One method works for every equation: set $x = 0$ and solve for $y$.
Substitute $x = 0$ everywhere in the equation.
Simplify.
Solve for $y$. That value is the y-coordinate of the y intercept.
If solving leaves no real solution, the graph never reaches the y-axis and there is no y intercept. Everything else is a shortcut for special forms.
The Y Intercept in Different Equation Forms
The same "set $x = 0$" idea takes a familiar shape in each common form.
Form | Equation | Y intercept |
|---|---|---|
Slope-intercept (line) | $y = mx + b$ | Read off: $b$, at $(0, b)$ |
Standard (line) | $ax + by = c$ | Set $x = 0$: $y = \dfrac{c}{b}$ |
Point-slope (line) | $y - y_1 = m(x - x_1)$ | Set $x = 0$: $y = y_1 - m x_1$ |
Quadratic (parabola) | $y = ax^2 + bx + c$ | Set $x = 0$: $y = c$, the constant term |
General function | $y = f(x)$ | Compute $f(0)$ |
The friendliest is slope-intercept form, where the y intercept $b$ is already one of the two named numbers — which is exactly why it is called slope-intercept form.
The Y Intercept of a Parabola
For a quadratic $y = ax^2 + bx + c$, setting $x = 0$ wipes out the $ax^2$ and $bx$ terms and leaves $y = c$. So the y intercept of an upward- or downward-opening parabola is the point $(0, c)$, always equal to the constant term. "Read the constant term" is the universal shortcut for these. (A sideways parabola of the form $x = ay^2 + by + c$ can cross the y-axis zero, one, or two times instead.)
Examples of the Y Intercept
With the definition and the universal method in place, here is the concept doing real work. The problems build from a one-line read up to fitting a parabola.
Example 1: Find the y intercept of the line $y = 4x - 7$.
The equation is in slope-intercept form $y = mx + b$ with $b = -7$, so the y intercept is the constant.
Final answer: $(0, -7)$, or $y = -7$.
Example 2: Find the y intercept of the line $3x + 5y = 15$.
A common first move is to read the y intercept straight off as 15 or 5, treating standard form like slope-intercept form. Test it by the definition: set $x = 0$, giving $3(0) + 5y = 15$, so $5y = 15$ and $y = 3$. The intercept is 3, not 15 or 5; the slip is forgetting that the y intercept is always found by setting $x = 0$, whatever the equation looks like.
Done correctly:
$$3(0) + 5y = 15 ;\Rightarrow; 5y = 15 ;\Rightarrow; y = 3.$$
Final answer: $(0, 3)$, or $y = 3$.
Example 3: Find the y intercept of the parabola $y = 2x^2 - 5x + 6$.
Set $x = 0$: $y = 2(0)^2 - 5(0) + 6 = 6$. The y intercept is the constant term.
Final answer: $(0, 6)$, or $y = 6$.
Example 4: Find the y intercept of the line through $(0, 4)$ and $(2, 0)$.
One of the given points already has $x = 0$. That point, $(0, 4)$, sits on the y-axis, so it is the y intercept directly.
Final answer: $(0, 4)$, or $y = 4$.
Example 5: Does the vertical line $x = 3$ have a y intercept?
A vertical line $x = 3$ runs parallel to the y-axis and never crosses it, so setting $x = 0$ contradicts $x = 3$. There is no y intercept.
Final answer: no y intercept.
Example 6: A parabola passes through $(1, 2)$, $(2, 5)$, and $(3, 14)$. Find its y intercept.
Write $y = ax^2 + bx + c$; the y intercept is $c$. Substituting the three points gives $a + b + c = 2$, $4a + 2b + c = 5$, and $9a + 3b + c = 14$. Subtracting the first from the second gives $3a + b = 3$; subtracting the second from the third gives $5a + b = 9$. Subtracting those gives $2a = 6$, so $a = 3$, then $b = -6$, then $c = 5$.
Final answer: $(0, 5)$, or $y = 5$. (Check: $y = 3(0)^2 - 6(0) + 5 = 5$.)
Where the Y Intercept Shows Up
The y intercept is the value of a quantity "at zero", so it carries plain meaning in nearly every applied setting.
Cost equations. The y intercept of cost-versus-units is the fixed cost: rent and salaries owed before a single unit is made.
Motion. The y intercept of a position-versus-time graph is the starting position, where the object sat at time zero.
Population growth. The y intercept of a population-versus-time graph is the initial population.
Regression lines. A regression line's y intercept is the predicted value when $x = 0$, meaningful only when $x = 0$ sits inside the data range.
The destination this points toward is calculus's tangent-line approximation: the linearization of a function near $x = 0$ is a line whose y intercept is the function's own value $f(0)$, and every Taylor series begins with that value.
Where Students Trip Up on the Y Intercept
Mistake 1: Reading a standard-form constant as the y intercept
Where it slips in: A line is given as $ax + by = c$, and the student calls $c$ the y intercept.
Don't do this: Treat the constant on the right as the intercept.
The correct way: Set $x = 0$ and solve. For $ax + by = c$ that gives $y = \tfrac{c}{b}$, not $c$.
Mistake 2: Confusing the y intercept with the x intercept
Where it slips in: Both are crossing points, and students swap the substitutions.
Don't do this: Find the y intercept by setting $y = 0$.
The correct way: The y intercept crosses the y-axis, so set $x = 0$. The x intercept crosses the x-axis, so set $y = 0$. Different point, different substitution.
Mistake 3: Forcing a y intercept onto a graph that has none
Where it slips in: A vertical line or a function undefined at $x = 0$ never meets the y-axis, yet a student invents one.
Don't do this: Apply the method blindly.
The correct way: Check whether the graph actually reaches the y-axis. For $y = \tfrac{1}{x}$, setting $x = 0$ gives $\tfrac{1}{0}$, undefined — no y intercept.
Key Takeaways
The y intercept is the point where a graph crosses the y-axis, always of the form $(0, c)$.
The universal method is to set $x = 0$ and solve for $y$.
For a line $y = mx + b$, the y intercept is $b$; for a parabola $y = ax^2 + bx + c$, it is the constant $c$.
In applied models the y intercept is the starting value, fixed cost, or initial condition.
The most common slip is reading a standard-form constant as the y intercept; set $x = 0$ instead.
Practice These Problems to Solidify Your Understanding
Find the y intercept of the line $y = -2x + 11$.
Find the y intercept of the line $4x - 3y = 18$.
Find the y intercept of the parabola $y = x^2 + 3x - 4$.
Answer to Question 1: $(0, 11)$. Answer to Question 2: $(0, -6)$. Answer to Question 3: $(0, -4)$. If Question 2 gave 18, check that you set $x = 0$ and solved for $y$ rather than reading the constant (see Mistake 1).
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