How to Find the X Intersection of a Function with the Horizontal Axis

X

This article was co-written by David Jia. David Jia is a tutoring teacher and founder of LA Math Tutoring, a private tutoring facility based in Los Angeles, California. With over 10 years of teaching experience, David teaches a wide variety of subjects to students of all ages and grades, as well as college admissions counseling and prep for SAT, ACT, ISEE, etc. scoring 800 in math and 690 in English on the SAT, David was awarded a Dickinson Scholarship to the University of Miami, where he graduated with a bachelor’s degree in business administration. Additionally, David has worked as an instructor in online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

There are 7 references cited in this article that you can see at the bottom of the page.

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In algebra, a two-dimensional coordinate graph has a horizontal horizontal axis, also known as the x-axis, and a vertical vertical axis, also called the y-axis. Where lines representing a series of values intersect these axes is called intersection. The y-intersection of the function with the vertical axis is the point where the line intersects the y-axis, and the x-intersection of the function with the horizontal axis is where the line intersects the x-horizontal axis. For simple problems, it is easy to find the x-intersection of the function with the horizontal axis by looking at the graph. You can find the exact intersection by solving math problems using the equation of the line.

Table of Contents

Using line graphs

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Determine the x-axis. The coordinate graph will have both x-axis and y-axis. The x-axis is a horizontal line (the line starts from left to right). The y-axis is the vertical line (the straight line going up and down). ^{[1] X Research Source} It is important that you look at the x-axis when determining the x intersection.

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Find the point of intersection of the line with the x-axis. This is the x intersection. ^{[2] X Research Source} If you are asked to find the x-intersection based on a graph, this will usually be the exact number (for example, at point 4). Usually, however, you will have to estimate using this method (for example, the point is between 4 and 5).

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Write down the pair of values for the x intersection. The value pair is written in the form

(

x

,

y

)

{displaystyle (x,y)}

(x,y)

and gives you the coordinates of the intersection. ^{[3] X Research Source} The first number of the pair of values is the intersection where the line intersects the x-axis (the x-intersection of the function with the horizontal axis). The second number will always be 0, because on the x-axis there will be no y value. ^{[4] X Research Sources}

For example, if the line intersects the x-axis at point 4, the pair of values for the x-intersection of the function with the x-axis is $($ $4$ $,$ $0$ $)$ ${displaystyle(4,0)}$ $(4,0)$ .

Using the equation of the line

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Determine that the equation of the line is in standard form. The standard form of a linear equation is

A

x

+

REMOVE

y

=

OLD

{displaystyle Ax+By=C}

Ax+By=C

. ^{[5] X Research Source} In this form,

A

{displaystyle A}

A

,

REMOVE

{displaystyle B}

REMOVE

, and

OLD

{displaystyle C}

OLD

is an integer,

x

{displaystyle x}

x

and

y

{displaystyle y}

y

are the coordinates of the point of intersection on the line.

For example, you might have the equation $2$ $x$ $+$ $3$ $y$ $=$ $6$ ${displaystyle 2x+3y=6}$ $2x+3y=6$ .

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Put $y$ ${displaystyle y}$ $y$ is 0. The x-intersection of the function with the horizontal axis is the intersection point of the line and the x-axis. ^{[6] X Research Source} At this point, the value of

y

{displaystyle y}

y

will be 0. ^{[7] X Research Source} So to be able to find the x intersection of the function with the horizontal axis, you need to put

y

{displaystyle y}

y

is 0 and solve for

x

{displaystyle x}

x

.

For example, if you substitute 0 for $y$ ${displaystyle y}$ $y$ , your equation will take the form: $2$ $x$ $+$ $3$ $($ $0$ $)$ $=$ $6$ ${displaystyle 2x+3(0)=6}$ $2x+3(0)=6$ , simplified would be $2$ $x$ $=$ $6$ ${displaystyle 2x=6}$ $2x=6$ .

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Solve the equation to find $x$ ${displaystyle x}$ $x$ . To do this, you need to isolate the variable x by dividing both sides of the equation by the coefficient. This method will give you the value of

x

{displaystyle x}

x

When

y

=

0

{displaystyle y=0}

y=0

, and this is the x-intersection of the function with the horizontal axis.

For example:
$2$ $x$ $=$ $6$ ${displaystyle 2x=6}$ $2x=6$
$2$ $x$ $2$ $=$ $6$ $2$ ${displaystyle {frac {2x}{2}}={frac {6}{2}}}$ ${frac {2x}{2}}={frac {6}{2}}$
$x$ $=$ $3$ ${displaystyle x=3}$ $x=3$

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Write down the value pair. You should remember that value pairs are written in the form

(

x

,

y

)

{displaystyle (x,y)}

(x,y)

. For the x intersection, the value of

x

{displaystyle x}

x

will be the value you calculated before, and the value

y

{displaystyle y}

y

will be 0, because

y

{displaystyle y}

y

is always zero at the x-intersection of the function with the horizontal axis. ^{[8] X Research Sources}

For example, for the line $2$ $x$ $+$ $3$ $y$ $=$ $6$ ${displaystyle 2x+3y=6}$ $2x+3y=6$ , the intersection x will be at the point $($ $3$ $,$ $0$ $)$ ${displaystyle(3,0)}$ $(3.0)$ .

Using quadratic equation

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Determine that the coordinates of the line are quadratic equations. A quadratic equation is an equation of the form

a

x

2

+

b

x

+

c

=

0

{displaystyle ax^{2}+bx+c=0}

ax^{{2}}+bx+c=0

. ^{[9] X Research Source} It has two solutions, which means that the line written in this form is a parabp and will have two intersections with the horizontal axis. ^{[10] X Research Source}

For example, the equation $x$ $2$ $+$ $3$ $x$ $-$ $ten$ $=$ $0$ ${displaystyle x^{2}+3x-10=0}$ $x^{{2}}+3x-10=0$ is a quadratic equation, so this line will have two intersections with the horizontal axis.

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Set up the formula for the quadratic equation. The formula is

x

=

-

b

\pm

b

2

-

4

a

c

2

a

{displaystyle x={frac {-bpm {sqrt {b^{2}-4ac}}}{2a}}}

x={frac {-bpm {sqrt {b^{{2}}-4ac}}}{2a}}

, in there

a

{displaystyle a}

a

is equal to the coefficient of the quadratic solution (

x

2

{displaystyle x^{2}}

x^{{2}}

),

b

{displaystyle b}

b

is equal to the variable of the first-order solution (

x

{displaystyle x}

x

), and

c

{displaystyle c}

c

is a constant. ^{[11] X Research Source}

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Substitute all values into the quadratic formula. Make sure you substitute the correct value for each variable of the line equation.

For example, if the equation of the line is $x$ $2$ $+$ $3$ $x$ $-$ $ten$ $=$ $0$ ${displaystyle x^{2}+3x-10=0}$ $x^{{2}}+3x-10=0$ , your quadratic formula will take the form: $x$ $=$ $-$ $3$ $\pm$ $3$ $2$ $-$ $4$ $($ $first$ $)$ $($ $-$ $ten$ $)$ $2$ $($ $first$ $)$ ${displaystyle x={frac {-3pm {sqrt {3^{2}-4(1)(-10)}}}{2(1)}}}$ $x={frac {-3pm {sqrt {3^{{2}}-4(1)(-10)}}}{2(1)}}$ .

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Simplify the equation. To do this, you need to complete all the multiplications first. Remember to pay attention to all signs of positive and negative numbers.

For example:
$x$ $=$ $-$ $3$ $\pm$ $3$ $2$ $-$ $4$ $($ $-$ $ten$ $)$ $2$ $($ $first$ $)$ ${displaystyle x={frac {-3pm {sqrt {3^{2}-4(-10)}}}{2(1)}}}$ $x={frac {-3pm {sqrt {3^{{2}}-4(-10)}}}{2(1)}}$
$x$ $=$ $-$ $3$ $\pm$ $3$ $2$ $+$ $40$ $2$ ${displaystyle x={frac {-3pm {sqrt {3^{2}+40}}}{2}}}$ $x={frac {-3pm {sqrt {3^{{2}}+40}}}{2}}$

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Calculate the exponent. Squared test

b

{displaystyle b}

b

. Then add it to the remaining number below the square root sign.

For example:
$x$ $=$ $-$ $3$ $\pm$ $3$ $2$ $+$ $40$ $2$ ${displaystyle x={frac {-3pm {sqrt {3^{2}+40}}}{2}}}$ $x={frac {-3pm {sqrt {3^{{2}}+40}}}{2}}$
$x$ $=$ $-$ $3$ $\pm$ $9$ $+$ $40$ $2$ ${displaystyle x={frac {-3pm {sqrt {9+40}}}{2}}}$ $x={frac {-3pm {sqrt {9+40}}}{2}}$
$x$ $=$ $-$ $3$ $\pm$ $49$ $2$ ${displaystyle x={frac {-3pm {sqrt {49}}}{2}}}$ $x={frac {-3pm {sqrt {49}}}{2}}$

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Solve the addition formula. Since the square root formula has

\pm

{displaystyle pm }

pm

, you need to do an addition problem, and a subtraction problem. Solving the addition problem will help you find the value

x

{displaystyle x}

x

.

For example:
$x$ $=$ $-$ $3$ $+$ $49$ $2$ ${displaystyle x={frac {-3+{sqrt {49}}}{2}}}$ $x={frac {-3+{sqrt {49}}}{2}}$
$x$ $=$ $-$ $3$ $+$ $7$ $2$ ${displaystyle x={frac {-3+7}{2}}}$ $x={frac {-3+7}{2}}$
$x$ $=$ $4$ $2$ ${displaystyle x={frac {4}{2}}}$ $x={frac {4}{2}}$
$x$ $=$ $2$ ${displaystyle x=2}$ $x=2$

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Solve the subtraction formula. It will give you the second value of

x

{displaystyle x}

x

. First, calculate the square root, then, find the difference in the numerator. Finally, divide it by 2.

For example:
$x$ $=$ $-$ $3$ $-$ $49$ $2$ ${displaystyle x={frac {-3-{sqrt {49}}}{2}}}$ $x={frac {-3-{sqrt {49}}}{2}}$
$x$ $=$ $-$ $3$ $-$ $7$ $2$ ${displaystyle x={frac {-3-7}{2}}}$ $x={frac {-3-7}{2}}$
$x$ $=$ $-$ $ten$ $2$ ${displaystyle x={frac {-10}{2}}}$ $x={frac {-10}{2}}$
$x$ $=$ $-$ $5$ ${displaystyle x=-5}$ $x=-5$

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Find the pair of values for the x-intersection of the function with the horizontal axis. You should remember that the value pair will have the x coordinate first, followed by the y . coordinate

(

x

,

y

)

{displaystyle (x,y)}

(x,y)

. Value

x

{displaystyle x}

x

will be the value that you calculated using the square root formula. Value

y

{displaystyle y}

y

will still be 0, because at the intersection of x with the horizontal axis,

y

{displaystyle y}

y

will always be 0. ^{[12] X Research Source}

For example, for the line $x$ $2$ $+$ $3$ $x$ $-$ $ten$ $=$ $0$ ${displaystyle x^{2}+3x-10=0}$ $x^{{2}}+3x-10=0$ , the x-intersection of the function with the horizontal axis lies at the point $($ $2$ $,$ $0$ $)$ ${displaystyle(2,0)}$ $(2.0)$ and $($ $-$ $5$ $,$ $0$ $)$ ${displaystyle (-5,0)}$ $(-5.0)$ .

Advice

If using the equation $y$ $=$ $m$ $x$ $+$ $b$ ${displaystyle y=mx+b}$ $y=mx+b$ , you need to know the slope of the line and the y-intersection of the function with the vertical axis. In the equation, m = slope of the line and b = the y-intersection of the function with the vertical axis. Set y equal to 0, and solve for x. You will find the x-intersection of the function with the horizontal axis.

Steps

Using line graphs

Using the equation of the line

Using quadratic equation

Advice

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