y-intercept

To graph any function that is of the form \(y=f(x)\) finding the intercepts is really important.

  • x-intercept: A point or the points where the graph intersects the x-axis
  • y-intercept: A point or the points where the graph intersects the y-axis

The points where the graph intersects the x-axis are x-intercepts. The points where the graph intersects the y-axis are y-intercepts

Let us learn more about y-intercept, the definition of y-intercept and the meaning of y-intercept, definition of y-intercept, example of y-intercept, and explore y-intercept calculator.

Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Lesson Plan 

What is y-Intercept? 

Meaning of y-Intercept

The y-intercept of a graph is (are) the point(s) 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.

The  y-intercept of y equals x plus 3 is (0, 3) whose x-coordinate is 0

The y-intercept of the graph is (0, 3) (or) 3 which can be obtained by substituting \(x=0\) in \(y=x+3\)

Thus, the y-intercept of any graph represented by an equation is obtained by substituting \(x=0\) in it and solving for \(y.\)


y-Intercept of a Straight Line

General Form

The equation of a line in general form is \[ax+by+c=0\]

For y-intercept, we substitute \(x=0\) and solve it for \(y.\)

\[ \begin{align} a(0)+by+c &=0\\[0.1cm] by+c&=0\\[0.1cm] y &= \dfrac{-c}{b} \end{align} \]

Thus, the y-intercept of the equation of a line in general form is:

\(\left(0,  -\dfrac{c}{b} \right)\)
or
\(\dfrac{-c}{b}\)

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.

You try by substituting \(x=0\) in \(y=mx+b\) and see whether you get \(b\) as the y-intercept.

Thus, the y-intercept of the equation of a line in the slope-intercept form is:

\(\left(0,  b \right)\)
or
\(b\)

Point-Slope Form

The equation of the line in the point-slope form is, \[y-y_1=m(x-x_1)\]

For y-intercept, we substitute \(x=0\) and solve it for \(y\)

\[ \begin{align} y-y_1 &= m (0-x_1)\\[0.1cm]
y-y_1&= -mx_1\\[0.1cm]
y&= y_1-mx_1
\end{align} \]

Thus, the y-intercept of the equation of a line in the point-slope form is:

\(\left(0,  y_1-mx_1 \right)\)
or
\(y_1-mx_1\)
 
important notes to remember
Important Notes
  1. We substitute \(x=0\) and solve for \(y\) to find the y-intercept.
    In the same way, we substitute \(y=0\) and solve for \(x\) to find the x-intercept.
  2. The lines parallel to the y-axis cannot have y-intercepts as they do not intersect the y-axis anywhere.
  3. The y-intercept of a line is widely used as an initial point while graphing a line by plotting two points.

How To Find y-Intercept of a Straight Line?

We have derived the formulas to find the y-intercept of a line where the equation of line was in different forms.

In fact, 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 \((0, y)\).

Examples

Equation of Line

Substitute \(x=0\) and solve for \(y\)

y-Interpcet

\(3x+5y -6=0\)

\( \begin{align} 3(0)+5y-6 &=0\\[0.1cm] 5y-6&=0\\[0.1cm] y &= \dfrac{6}{5} \end{align} \) \(\dfrac{6}{5}\) (or) \(\left( 0, \dfrac{6}{5}\right)\)

\( y= 2x-3\)

\(y=2(0)-3=-3\) -3 (or) (0, -3)

The y- intercept of the second equation of the table is shown in the graph below.

y-intercepy of y equals 2 x minus 3 is (0, minus 3)


How To Find y-intercept of a Quadratic Function or a Parabola?

The procedure for finding the y-intercept of a quadratic function (parabola) is the same as that of a line (as discussed in the previous section).

Examples

Equation of Parabola

Substitute
\(x=0\)
and 
solve for \(y\)

y-Intercept

\(y= x^2 -2x -3\)

\(y=0^2-2(0)-3=-3\) -3 (or) (0, -3)

\( y= 2x^2+5x-3\)

\(y=2(0)^2+5(0)-3=-3\) -3 (or) (0, -3)

The y intercept of y equals x squared minus 2 x minus 3 is (0, minus 3)


y-Intercept Calculator

Here is the y-intercept calculator.

You can enter any function of the form \(y=f(x)\), and the calculator will show its y-intercept.

 
tips and tricks
Tips and Tricks
  1. The y-intercept of the polynomial function of the form \(y=a_1x^n+a_2 x^{n-1}+...+a_n\) is just its constant term \(a_n\) (or) \((0, a_n)\).
  2. The y-intercept of a function can be easily found by graphing it using the graphing calculator and locating the point where the graph cuts the y-axis.
  3. A function has only one y-intercept because otherwise, it fails the vertical line test.

Solved Examples

Example 1

 

 

Can we help James find the y-intercept of the graph of \(y= x+ \dfrac{1}{x}\)?

Solution

To find the y-intercept of the given graph, we substitute \(x=0\).

Then we get:

\[ y=0+ \dfrac 1 0 = \infty\]

Hence the given graph has no y-intercept. It means the graph does not intersect the y-axis anywhere.

The y-intercept of the graph does NOT exist
Example 2

 

 

Mia is asked to find the intercepts of the graph represented by the equation \(x=y^{2}+2 y-3\). Can we help her?

Solution

x-Intercept

To find the x-intercept, we substitute \(y=0\) in the given equation and solve for \(x\). Then, \[ x=0^2+2(0)-3=-3\]

y-Intercept

To find the y-intercept, we substitute \(x=0\) in the given equation and solve for \(y\). Then,

\[\begin{align}  y^2+2y-3 &=0\\[0.2cm]
(y+3)(y-1)&=0\\[0.2cm]
y=-3;\;\; &y=1 \end{align} \]

Thus,

x-intercept is (-3, 0)
y-intercepts are (0, -3) and (0, 1)

Interactive Questions

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

We hope you enjoyed learning about y-Intercept with the simulations and practice questions. Now, you will be able to easily solve problems on the meaning of y-intercept, definition of y-intercept, example of y-intercept, and y-intercept calculator.

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Frequently Asked Questions (FAQs)

1. How do you find the y-intercept on the graphing calculator?

We use the button "\(y=\)" on the calculator to enter the function and then press on the "graph."

Then the graph of the function is displayed on the screen.

Then press on the "zoom" button.

Then press on the "trace" button and enter 0

It will then show the value of \(y\) at which \(x\) is 0

2. What is the y-intercept of \(y= 3x?\)

To find the y-intercept, substitute \(x=0\)

Then \[y=3(0)=0\]

So the y-intercept of \(y=3x\) is \((0,0)\)

3. What is the slope of \(y = 6?\)

Comparing \(y=6\) with \(y=mx+b\), we get, \[m=0\\b=6\]

Here, \(m\) is the slope of the line.

Thus, the slope of \(y=6\) is 0

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