Y Intercept: Definition, Formula & Examples
Y Intercept: Definition, Formula & Examples
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=c/b |
| Point-slope (line) | y−y1=m(x−x1) | Set x=0: y=y1−mx1 |
| Quadratic (parabola) | y=ax²+bx+c | Set x=0: y=c |
| 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=ax²+bx+c, setting x=0 wipes out the ax² 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=ay²+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.
Example 1: Find the y intercept of the line y=4x−7.
The equation is in slope-intercept form 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.
Set x=0, giving 5y=15 and y=3. The intercept is 3.
Final answer: (0,3), or y=3.
Example 3: Find the y intercept of the parabola y=2x²−5x+6.
Set x=0: y=6.
Final answer: (0,6), or y=6.
Example 4: Find the y intercept of the line through (0,4) and (2,0).
The 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 runs parallel to the y-axis and never crosses it, so 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.
The y intercept is c from the form y=ax²+bx+c.
Final answer: (0,5), or y=5.
Where the Y Intercept Shows Up
The y intercept 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.
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.
The correct way: Set x=0 and solve.
Mistake 2: Confusing the y intercept with the x intercept
Where it slips in: Students swap the substitutions.
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.
Mistake 3: Forcing a y intercept onto a graph that has none
Where it slips in: A vertical line never meets the y-axis, yet a student invents one.
The correct way: Check whether the graph actually reaches the y-axis.
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=ax²+bx+c, it is the constant c.
- 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=x²+3x−4.