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.

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 \(3x+(-2y)=12\).
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{(0,-1)}\)
  2. \(\color{blue}{-5}\)
  3. \(\color{blue}{-6}\)

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