# 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.**  
1. Substitute x=0 everywhere in the equation.  
2. Simplify.  
3. 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  
1. Find the y intercept of the line y=−2x+11.  
2. Find the y intercept of the line 4x−3y=18.  
3. Find the y intercept of the parabola y=x²+3x−4.