x intercept

x intercept

Tim and Janne were observing various graphs. They observed that some crossed the x-axis, while others didn't. They were curious to know more about the curves and lines that cross the x-axis.

Can we say that x-intercept is the value of the \(x\) coordinate of a point where the value of \(y\) coordinate equal to zero?

Let’s learn about x-intercept, x-intercept in an equation, x-intercept formula, x-intercept definition, and x-intercept example, and explore x-intercept calculator for better understanding.

Check out the interactive examples 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 x-Intercept?

The x-intercept for any curve is the value of x coordinate of the point where the graph cuts x-axis, or we can say that x-intercept is the value of the x coordinate of a point where the value of y coordinate equal to zero.


General Formulae for X-Intercept

Staright Line

The general form of a straight line is

\(ax + by +c = 0\)

where \(a, b, c\) are constants

x-intercept of the line can be obtained by putting \(y=0\)

\(x \text{-intercept} = \dfrac{-c}{a}\)

Slope Intercept Line

Slope-intercept form of a straight line is

\(y = mx + c\)

where \(m\) is the slope of the line
\(c\) is the y-intercept

The x-intercept of the line can be obtained by putting \(y=0\)

\(x \text{-intercept}= \dfrac{-c}{m}\)

Point Slope Line

Point-slope form of a straight line is 

\(y -b = m(x - a)\)

where \(m\) is the slope of the line
\((a, b)\) is a point on the line

The x-intercept of the line can be obtained by putting \(y=0\)

\(x \text{-intercept}= \dfrac{am -b}{m}\)


How to Find x-Intercept of a Straight Line?

To find the x-intercept of a line of the form \(y= mx +b\),
substitute y = 0

Example

To find the x-intercept of any line for example the line \(y = 2x -4\), put \(y=0\) in the equation of line.

Solution

\[\begin{align}
0 &= 2x - 4\\[0.2cm]
4 &= 2x\\[0.2cm]
2 &= x\\[0.2cm]
\end{align}\]

So, the x-intercept of the line \(y = 2x -4\) is \(2\)

Similarly, by substituting x = 0, we can find the y-intercept

x-intercept of a line

Let's see an x-intercept example.

Example

John wants to find the equation of a line with the slope equal to 2 and the x-intercept equal to -5. Can you help him?

Solution

The general equation of a line with slope m is \(y = mx + c\)
The x-intercept of the line is \(-5\)
\[\text{using the formula for x-intercept}\]
\[\begin{align}
x &= \dfrac{-c}{m} \\[0.2cm]
-5 &= \dfrac{-c}{2} \\[0.2cm]
10 &= c \\[0.2cm]
\end{align}\]
\[\text{Put the value of c and m}\]
\[y = 2x + 10 \]

Equation of the line is \(y=2x+10\)

Intercept Calculator

 

 
tips and tricks
Tips and Tricks
  • If the x-intercept and the y-intercept of a line are equal, then the line forms a 45 degree angle with the x-axis.
  • To find the x-intercept of any line ax+by+c=0, put the value of y as 0
  • To find the point of intersection of two lines, solve the two equations simultaneously.
  • For two perpendicular lines, the product of their slopes is -1

How to Find x-Intercept of a Quadratic Function or a Parabola?

The x-intercept of any curve can be obtained using a similar technique. We just have to put the value of y as 0 in the equation of the curve.

The general equation of a parabola is \(y = ax^2 + bx + c\). We can calculate x-intercept by puting y as 0

Example

Find x-intercept of the parabola \(y = x^2 - 3x +2\).

Solution

Put \(y=0\) in the equation of parabola.

\[0 = x^2 - 3x +2\]
We can apply regular methods to solve a quadratic equation and calculate the value of x-intercepts.
\(x = 2 \text{ and } x = 1\) are the two roots of the equation.
It means that the graph is crossing x-axis at two different points.

So, \(x = 2 \text{ and } x = 1\) are the two x-intercepts of the parabola

Similarly we can calculate y-intercept by putting x = 0

x and y intercept

 
important notes to remember
Important Notes
  • The x-intercept of a line is the distance of the x coordinate from the point where the line cuts the x-axis from the origin.
  • The x-intercept is a point on the graph where y is zero.
  • If a line is parallel to the y-axis, its x-intercept is not defined.
  • For any curve, the x-intercepts of the curve = roots of the equation = solution of the equation = zeros of the curve.

Solved Examples

Example 1

 

 

Shawn says that the line \(2y = 3x - 6\) has x-intercept equal to \(-3\) and y-intercept equal to \(2\) while Judy says that the line has x-intercept equal to \(2\) and y-intercept equal to \(-3\).
Who is right?

Solution

Given the line \(2y = 3x - 6\)
x-intercept can be calculated by putting y = 0
\[\begin{align}
2 \times 0 &= 3x -6\\[0.2cm]
0 &= 3x -6\\[0.2cm]
6 &= 3x\\[0.2cm]
2 &= x \\[0.2cm]
\text{x intecept of the line is 2}\\[0.2cm]
\end{align}\]
y-intercept can be calculated by putting x = 0
\[\begin{align}
2y &= 3 \times 0 -6\\[0.2cm]
2y &= 0 -6\\[0.2cm]
2y &= -6\\[0.2cm]
y &= -3\\[0.2cm]
\text{y-intercept of the line is -3} \\[0.2cm]
\end{align}\]

\(\therefore\) Judy is right.
Example 2

 

 

Mathew draw a line with slope -2  which passes through point \(P(a, 1)\). If the sum of both the intercepts of the line is equal to 6, find the value of \(a\).

Solution

General equation for a line with a given slope and a point is
\[y - b = m(x - a)\]

Where \((a, b)\) is the point on the line

\[\begin{align}
y - 1 &= -2(x - a)\\[0.2cm]
\text{x-intercept} &= \dfrac{1}{2} + a\\[0.2cm]
\text{y-intercept} &= 1 + 2a\\[0.2cm]
\text{sum of intercepts} &= 1 + 2a + \dfrac{1}{2} + a\\[0.2cm]
\text{sum of intercepts} &= 3a + \dfrac{3}{2}\\[0.2cm]
6 &= 3a + \dfrac{3}{2}\\[0.2cm]
\dfrac{9}{2} &= 3a\\[0.2cm]
\dfrac{3}{2} &= a\\[0.2cm]
\end{align}\]

\(\therefore\) Value of \(a\) is \(\dfrac{3}{2}\)
Example 3

 

 

Find the x and y-intercepts of the curve \(9x^2 + 16y^2 = 25\)

Solution

x-intercept of the curve can be calculated by putting \(y =0\)
\[\begin{align}
9x^2 + 16 \times 0^2 &= 25\\[0.2cm]
9x^2 + 0 &= 25\\[0.2cm]
x^2  &= \dfrac{25}{9}\\[0.2cm]
x^2  &= (\dfrac{5}{3})^2\\[0.2cm]
x  &= \pm\dfrac{5}{3}\\[0.2cm]
\end{align}\]
y-intercept of the curve can be calculated by putting \(x =0\)
\[\begin{align}
9 \times 0^2 + 16y^2 &= 25\\[0.2cm]
0 + 16y^2 &= 25\\[0.2cm]
y^2 &= \dfrac{25}{16}\\[0.2cm]
y^2  &= (\dfrac{5}{4})^2\\[0.2cm]
y  &= \pm\dfrac{5}{4}\\[0.2cm]
\end{align}\]

\(\therefore\) x-intercepts are \(\dfrac{5}{3} \text{ and} -\dfrac{5}{3}\)
y-intercepts are \(\dfrac{5}{4} \text{ and} -\dfrac{5}{4}\)

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 x-intercept with the simulations and practice questions. Now you will be able to easily solve problems on x-intercept in an equation, x-intercept formula, x-intercept definition, and x-intercept example.

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Frequently Asked Questions (FAQs)

1. Can 0 be an x-intercept?

Yes, 0 can be an x-intercept for the line \(y = mx\), where m is the slope of the line.

2. What is the y-intercept formula?

y-intercept formula for a line \(y = mx + c\) is \(y = c\).

3. How do you find x in an equation?

We can find \(x\) by putting the value of \(y = 0\) in any equation. The value of \(x\) so obtained is the x-intercept of the curve.

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