Home » Math Vocabulary » What is Y Intercept? Definition, Formula, Equation, Examples

- What Is Y-Intercept?
- Y-Intercept Formula
- Y-Intercept of a Straight Line
- Solved Examples on Y-Intercept
- Practice Problems on Y-Intercept
- Frequently Asked Questions about Y-intercept

## What Is Y-Intercept?

**The y-intercept of a graph is the point where the graph crosses the y-axis.**

Since the y-intercept is a point on the y-axis, its x-coordinate is always 0. Thus, the coordinates of the y-intercept are of the form (0, y).

Take a look at the y-intercept of a line on a graph shown** **below.

**Note:**

If the graph is the graph of a function, then it has at most one y-intercept. According to the vertical line test, a graph represents a function if and only if it crosses every vertical line only once. Considering the y-axis as one such vertical line, the graph of a function cannot have more than one y-intercept.

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## Y-Intercept Definition

The point at which the graph function crosses the y-axis is called the y-intercept.

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## Y-Intercept Formula

The y intercept is of the form of (0, y). So, to find the y-intercept of a function, substitute x = 0 and solve for y.

**Y-intercept example**:

Let the equation of a line be y=10x+4.

Substitute x = 0.

y = 10(0) + 4

y = 0 + 4

y = 4

Thus, the y-intercept of the line is (0, 4).

## Y-Intercept of a Straight Line

The y-intercept of a straight line is the point where the line intersects the y-axis. A line has only one y-intercept.

**Finding the y-Intercept of a Line using General Form**

The general form of the equation of the straight line is **ax+by+c=0**

To find y-intercept, substitute x = 0 and solve for y.

a(0)+ by + c =0

0+ by + c =0

by = -c

$y = \;-\;\frac{c}{b}$

The y-intercept of a line **ax + by + c = 0** is (0, $\frac{-c}{b}$) or simply $\frac{-c}{b}$.

**Finding the y-Intercept of a Line using Slope-Intercept Form**

The slope-intercept form of the equation of the line is** ****y = mx + b**, where m is the slope and b is the y-intercept of the line.

To find y-intercept, substitute **x** = 0 and solve for y.

y = mx + b

y = m(0) + b

y = b

The y-intercept of the equation of a line in slope-intercept form is (0, b) or b.

**Finding the y-Intercept of a Line using Point-Slope Form**

The point-slope form of the equation of the line is given by (y – y_{1}) = m(x – x_{1}), where m is the slope of the line and (x_{1}, y_{1}) is a point on the line.

To find y-intercept, substitute **x**** **= 0 and solve for y.

y- y_{1} = m(0 – x_{1})

y – y_{1} = – mx_{1}

y = – mx_{1} – y_{1}

y = y_{1} – mx_{1}

The y-intercept of the a line in point-slope (y – y_{1}) = m(x – x_{1}) form is (0, y_{1} – mx_{1}) or**(y _{1}**

**– mx**.

_{1})## Y-Intercept on a Graph

The point where the graph crosses the y-axis is the y-intercept.

**Example:** The line intersects the y-axis at (0, 3). Thus, the y-intercept is (0, 3) or 3.

## Y-Intercept of a Quadratic Function (Parabola)

The quadratic function has the standard form y = ax^{2} + bx + c. The y-intercept is always equal to the value of c.

To find the y-intercept, we just have to substitute x = 0 in the equation and solve for y. Thus, the corresponding y-intercept will be y or (0, y).

**Example:** y = x^{2} – 2x – 3

Substitute x = 0 and solve for y.

y=0 – 2(0) – 3

y = – 3

Thus, the y-intercept is -3 or (0, -3).

## Facts on Y-Intercept

- To find the y-intercept, substitute x = 0 and solve for y.
- The lines parallel to the y-axis do not have the y-intercept as they don’t intersect with the y-axis.
- A function cannot have more than one y-intercept.
- The y-intercept of a polynomial function in the form y = a
_{1}x_{n}+ a_{2}x_{n-1}+ …+ a_{n}is its constant term a_{n}or (0, a_{n}).

## Conclusion

In this article, we have learned about the y-intercept,** **definition, formula, and its different form with examples. Now, let us solve a few examples and practice problems on y-intercept.

## Solved Examples on Y-Intercept

**Example 1: Identify the y-intercept of the line ****y = 2x – 4**

**Solution:**

To find y-intercept, substitute x = 0 and solve for y.

y = 2(0) – 12

y = -12

Thus, the y-intercept of the line y = 2x – 4 is (0,- 12).

**Example 2: Find the y-intercept of the following quadratic function: y = x ^{2} – 6x + 4**

**Solution:**

To find y-intercept, substitute **x** = 0 and solve for y.

y = 0^{2} **–** 6(0) + 4

y = 4

Thus, the y-intercept of the equation **y = x ^{2} – 6x + 4** is (0, 4).

**Example 3: If (0,-3) is the y-intercept of the function y = 3x ^{2} + ax + b, then find the value of b.**

**Solution:**

Equation: y = 3x^{2} + a**x** + b

Y-intercept =** **(0,**–**3)

b = ?

Using the y-intercept formula, substitute **x**** **= 0 in the given equation,

y = 3x^{2} + ax + b

y = 3(0)^{2} + a(0) + b

y = b

Thus, b = **–**3

## Practice Problems on Y-Intercept

1

### Find the y-intercept of the line y = x?

1

-1

Not defined

CorrectIncorrect

Correct answer is: 0

The y-intercept of the line y = x is 0.

y $= \frac{(0^{2}\;-\;1)}{0}$

y $= \frac{-\;1}{0}$, which is not defined.

2

### If the y-intercept of the line y = 5x + a is 5, then find the value of a.

3

4

5

6

CorrectIncorrect

Correct answer is: 5

Substitute x = 0 in y = 5x + a.

y = 5(0) + a

y = a

The y-intercept is a = 5.

3

### What is the y-intercept of the line whose equation is 4x + 6y = 10?

$\frac{2}{3}$

$\frac{5}{3}$

$\frac{3}{2}$

$\frac{3}{5}$

CorrectIncorrect

Correct answer is: $\frac{5}{3}$

Put x = 0 in 4x + 6y = 10.

0 + 6y = 10

y $= \frac{5}{3}$

## Frequently Asked Questions about Y-intercept

**Does a vertical line have a y-intercept?**

No, a vertical line does not have a y-intercept.

**What is the x-intercept of a line?**

The x-intercept of a line is the point where the line crosses the x-axis.

**How do you find the y-intercept of a line?**

To find the y-intercept, set x equal to zero in the equation of the line and solve for y.

**Is the y-intercept always a whole number?**

No, the y-intercept can be any real number.