# How to Find the y-Intercept of a Line?

The $$y$$-intercept is the point where the graph intersects the $$y$$-axis. The following guide learn you how to find the $$y$$-intercept of a line.

An intercept of a function is a point where the graph of the function cuts the axis. The $$y$$-intercept of a function is a point where its graph would meet the $$y$$-axis.

## Step by step guide to finding the $$y$$-intercept

The $$y$$ intercept of a graph is the point where the graph intersects the $$y$$-axis. We know that the $$x$$-coordinate of any point on the $$y$$-axis is $$0$$. So the $$x$$-coordinate of a $$y$$-intercept is $$0$$.

Here is an example of a $$y$$-intercept. Consider the line $$y = x+ 3$$. This graph meets the $$y$$-axis at the point $$(0,3)$$. Thus $$(0,3)$$ is the $$y$$-intercept of the line $$y = x+ 3$$.

### $$y$$-intercept Formula

The $$y$$-intercept of a function is a point where its graph would meet the $$y$$-axis. The $$x$$-coordinate of any point on the $$y$$-axis is $$0$$ and we use this fact to derive the formula to find the y-intercept. i.e., the $$y$$-intercept of a function is of the form $$(0, y)$$. Thus, the formula to find the $$y$$-intercept is:

Here are the steps to find the $$y$$-intercept of a function $$y = f(x)$$,

• Substitute $$x = 0$$ in it.
• Solve for $$y$$.
• Represent the $$y$$-intercept as the point $$(0, y)$$.

### $$y$$-intercept of a straight line

A straight line can be horizontal or vertical or slanting. The $$y$$-intercept of a horizontal line with equation $$y = a$$ is $$(0, a)$$ and the $$y$$-intercept of a vertical line does not exist.

### $$y$$-intercept in general form

The equation of a straight line in general form is $$ax+by+c=0$$. For $$y$$-intercept, we substitute $$x=0$$ and solve it for $$y$$:

$$a(0)+by+c=0$$

$$by+c=0$$

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

Thus, the $$y$$-intercept of the equation of a line in general form is: $$(0, -\frac{c}{b})$$ or $$-\frac{c}{b}$$.

### $$y$$-intercept in slope-intercept form

The equation of the line in the slope-intercept form is, $$y=mx+b$$. By the definition of the slope-intercept form itself, $$b$$ is the $$y$$-intercept of the line. Thus, the $$y$$-intercept of the equation of a line in the slope-intercept form is $$(0, b)$$ or $$b$$.

### $$y$$-intercept in point-slope form

The equation of the line in the point-slope form is, $$y-y_1=m(x-x_1)$$. For the $$y$$-intercept, substitute $$x=0$$ and solve it for $$y$$:

$$y-y_1=m (0-x_1)$$

$$y-y_1=-mx_1$$

$$y=y_1-mx_1$$

Thus, the $$y$$-intercept of the equation of a line in the point-slope form is $$(0,y_1-mx_1)$$ or $$(y_1-mx_1)$$.

### How to find $$y$$-intercept?

We have derived formulas for finding the $$y$$-intercept of a line, which is the equation of a straight line in various forms. We do not need to apply any of these formulas to find the $$y$$-intercept of a straight line. We just substitute $$x=0$$ in the equation of the line and solve for $$y$$. Then the corresponding $$y$$-intercept is $$y$$ or $$(0, y)$$.

### $$y$$-intercept of a quadratic function (parabola)

The method for finding the $$y$$-intercept of a quadratic function or the $$y$$-intercept of a parabola is the same as that of a line. If a quadratic equation is given, substitute $$x = 0$$ and solve for $$y$$ to get the $$y$$ intercept.

### Finding the $$y$$-Intercept – Example 1:

Find the $$y$$-intercept of the equation $$y=x^2-2x-3$$.

Solution:

Substitute $$x=0$$ and solve for $$y$$:

$$y=0^2-2(0)-3$$

$$y=-3$$

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

## Exercises for Finding the $$y$$-Intercept

1. Find the $$y$$-intercept of the equation $$y=(x^2-1)$$.
2. If the $$y$$-intercept of the function $$y=3x^2+ax+b$$ is $$(0,-5)$$, find the value of $$b$$.
3. Find the $$y$$-intercept of the equation $$3x+(-2y)=12$$.
1. $$\color{blue}{(0,-1)}$$
2. $$\color{blue}{-5}$$
3. $$\color{blue}{-6}$$

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