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Identifying Intercepts of a Line

Identifying Intercepts of a Line

Author: Sophia Tutorial
Description:

Given the equation of a line, identify the x- and y- intercept.

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Tutorial
what's covered
This tutorial covers identifying intercepts of a line, through the exploration of:
  1. Defining x- and y-Intercepts
  2. Identifying Intercepts from a Graph
  3. Identifying Intercepts from an Equation


1. Defining x- and y-Intercepts

The x-intercept is the location on a graph where a line or a curve intersects the x-axis. The coordinate pair of any x-intercept is (x, 0) because the value of y is always 0 on the x-axis.

term to know
x-Intercept
The location on a graph where a line or curve intersects the x-axis: (x, 0)

The y-intercept is the location on the graph where a line or curve intersects the y-axis. The coordinate pair of any y-intercept is (0, y) because the value of x is always 0 on the y-axis.

term to know
y-Intercept
The location on a graph where a line or curve intersects the y-axis: (0, y)


2. Identifying Intercepts from a Graph

You can visually identify the x- and y-intercepts on a graph. For example, the graph below shows the height above ground of a plane during its descent in relation to time in minutes. The y-intercept is represented by the pair (0, 30,000). This means that after 0 minutes of descent, the plane is 30,000 feet above the ground.



The x-intercept of the graph is represented by the order paired (10, 0). This means that after 10 minutes, the height of the plane is 0 feet above ground.


3. Identifying Intercepts from an Equation

You can also find the x- and y-intercepts from an equation. An equation in two variables is an equation with terms involving two distinct variables. Most commonly, these variables are x and y.

term to know
Equation in Two Variables
An equation with terms involving two distinct variables

For example, the equation below represents the balance of debt during a 24-month repayment period, with the corresponding graph representation.

y equals negative 45 x plus 1080



You can find the x-intercept by substituting 0 for y in the equation and then solving the equation for x.



Doing this, your expression becomes:
0 equals negative 45 x plus 1080
You can start to solve this equation for x by subtracting 1,080 from both sides. Next, divide by -45 on both sides, which provides x equals 24. Therefore, the x-coordinate of the x-intercept is 24, and the ordered pair of the x-intercept is (24, 0). This means that after 24 months, the balance of debt is $0.

table attributes columnalign left end attributes row cell 0 equals negative 45 x plus 1080 end cell row cell 0 minus 1080 equals negative 45 x plus 1080 minus 1080 end cell row cell negative 1080 equals negative 45 x end cell row cell fraction numerator negative 1080 over denominator negative 45 end fraction equals fraction numerator negative 45 x over denominator negative 45 end fraction end cell row cell space space space space space space space space space space 24 equals x end cell end table
Similarly, you can find the y-intercept by substituting 0 for x in the equation and then solving the equation for y.



Substituting 0 for x, your equation is:
y equals negative 45 left parenthesis 0 right parenthesis plus 1080
Solve the equation by multiplying -45 times 0, which equals 0. Next, add 0 and 1,080, which equals 1,080. Therefore, y equals 1,080. The y-coordinate of the y-intercept is 1,080, and the ordered pair of the y-intercept is (0, 1080). This means that after 0 months, the balance of debt is $1,080.

table attributes columnalign left end attributes row cell y equals negative 45 left parenthesis 0 right parenthesis plus 1080 end cell row cell y equals 0 plus 1080 end cell row cell y equals 1080 end cell end table

big idea
Note that for equations written as y = ax + b (slope intercept form), as in the preceding example, the y-intercept can easily be defined by b, or the constant value at the end of the equation, since a times x will always be 0 when x is 0 for any value of a.
formula
y equals m x plus b
try it
Try finding the x- and y-intercepts for the following equation in slope intercept form.
y equals 5 x minus 30
To find the x-intercept, you know that the value of y will be 0, so you can substitute 0 for y into the equation and solve for x. This provides 0 equals 5x minus 30. To solve for x, start by adding 30 to both sides of the equation. Divide by 5 on both sides, which simplifies to 6 is equal to x. The x-intercept can be written as the ordered pair (6,0).

table attributes columnalign left end attributes row cell 0 equals 5 x minus 30 end cell row cell 0 plus 30 equals 5 x minus 30 plus 30 end cell row cell 30 equals 5 x end cell row cell 30 over 5 equals fraction numerator 5 x over denominator 5 end fraction end cell row cell space space space space space 6 equals x end cell row cell space space space space space left parenthesis 6 comma 0 right parenthesis end cell end table
hint
Notice that the y value of the ordered pair is 0, because the value of y is always 0 at the x-intercept.
Next, to find the y-intercept, substitute 0 for x in the equation and solve for y. You know that the y-intercept will always be the value of b in our equation, because when x is 0, a times x will always equal 0 for any value of a. However, you can complete the steps to solve the equation for y to show that this is true. Substitute 0 for x, then simplify the right side of the equation, arriving at the expression y equals -30. The y-intercept can be written as the ordered pair (0, -30).

table attributes columnalign left end attributes row cell y equals 5 left parenthesis 0 right parenthesis minus 30 end cell row cell y equals 0 minus 30 end cell row cell y equals negative 30 end cell row cell left parenthesis 0 comma negative 30 right parenthesis end cell end table
hint
Notice that the x value of the ordered pair is 0, because the value of x is always 0 at the y-intercept.

summary
Today you learned the definition of the x- and y-intercepts on a graph, which is where a line or curve intersects each respective axis. You also learned how to identify the x- and y-intercepts from a graph, as well as how to identify the x- and y-intercepts from an equation.

Source: This work is adapted from Sophia author Colleen Atakpu.

Terms to Know
Equation in Two Variables

An equation with terms involving two distinct variables.

X-Intercept

The location on a graph where a line or curve intersects the x-axis: (x, 0).

Y-Intercept

The location on a graph where a line or curve intersects the y-axis: (0, y).

Formulas to Know
Slope-Intercept Form of a Line

y equals m x plus b