Topics
4. Proportional Relationships & Lines
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United States of America
Grade 8

4.05 Identify the y-intercept

Lesson
Practice

Introduction

We have learned how to  identify the slope  of a line on a coordinate plane. We know that the equation of a line passing through the origin is y=mx. But what if the line passes through other points on the y-axis? We will now learn more about the graph of a line which includes the y-intecept.

The y-intercept

The y-intercept is the point where a graph crosses the y-axis. The y-intercept of the graph shown below is (0,2).

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-2
-1
1
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x
-3
-2
-1
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y

The x-coordinate of every y-intercept is 0. This is because if a point is located on the y-axis then it is always located at 0 on the x-axis. So if we are looking at a table, we simply need to find 0 in the x column. The associated value will give us our y.

What is the y-intercept of the table shown below?

xy
-24
-17
010
113

If we find 0 in the x column we see that the associated y value is 10. This means the y-intercept is (0,10). The x-coordinate is always 0 for the y-intercept, so we can just say that the y-intercept is 10.

Examples

Example 1

Consider the following table of values.

x-1012
y-5-214

Identify the coordinates of the y-intercept.

Worked Solution
Create a strategy

We can use the fact that the y-intercept is the point where the x-coordinate, is always 0.

Apply the idea

Based on the table, when the value of x is 0, the value of y is -2. This means that the y-intercept is at (0,-2).

Idea summary

The y-intercept is the point where a graph crosses the y-axis. The x-coordinate is always 0.

Slope and y-intercept from an equation

From previous lessons, we learned the equation of a line that passes through the origin is y=mx. For a line that passes through other points in the y-axis, the equation of the line is y=mx+b, where b is the y-intercept.

Given the equation of the line y=2x-4, the slope of the line is the coefficient of x which is 2 while the y-intercept is the constant number -4. The coordinates of the y-intercept are (0,-4).

The general slope-intercept form of a line, where m is the slope and b is the y-intercept is: y=mx+b

Exploration

Let us use the applet below to observe how the slope and y-intercept impact the graph of a line.

Loading interactive...

The applet above highlights that the m value affects the steepness of the line, or the slope.

  • If m < 0, the slope is negative and the line is decreasing.
  • if m > 0, the slope is positive and the line is increasing.
  • if m = 0 the slope is 0 and the line is horizontal.
  • Also, the farther the value of m is from 0 the steeper the line. t

We also found that the b value affects the y-intercept, or where the line crosses the y-axis.

  • If b is positive then the line crosses the y-axis above the origin.
  • If b is negative then the line crosses the y-axis below the origin.
  • If b is 0, or not in the equation at all, then the line crosses the y-axis at the origin.

Examples

Example 2

State the slope and y-intercept of the equation y=-4x+5.

Worked Solution
Create a strategy

We can use the general slope-intercept form: y=mx+b, where m is the slope and b is the \\y-intercept.

Apply the idea

The equation y=-4x+5 is in the form y=mx+b:

\displaystyle m\displaystyle =\displaystyle -4Identify the slope
\displaystyle b\displaystyle =\displaystyle 5Identify the y-intercept

Example 3

Consider the following graph of a lne:

-5
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x
-5
-4
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y
a

What is the slope of the line shown in the graph?

Worked Solution
Create a strategy

The slope of a line is equal to the vertical 'rise' divided the by horizontal 'run'. It is the ratio of the vertical change to the horizontal change.

Apply the idea

Considering two points from the graph: (0,1) and (5,0), from (0,1) requires moving 1 unit down and 5 units to the right.

The slope is \dfrac{-1}{5} or -\dfrac{1}{5}

b

What is the y- value of the y-intercept of the line shown in the graph?

Worked Solution
Create a strategy

The y-intercept is the point where the line intersects the y-axis.

Apply the idea
-5
-4
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-1
1
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5
x
-5
-4
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y

Looking at the graph, the line intersects the y-axis at point (0,1). Thus, the y-value of the y-intercept is y=1.

c

Write the equation of the line in slope-intercept form.

Worked Solution
Create a strategy

Substitute the values of the slope and y-intercept in the slope-intercept form of the equation of a line.

Apply the idea
\displaystyle y\displaystyle =\displaystyle mx+bSlope-intercept form
\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{5}x+1Substitute the values of the slope and y-intercept
Idea summary

The general slope-intercept form of a line:

\displaystyle y=mx + b
\bm{m}
is the slope.
\bm{b}
is the y-intercept.

Outcomes

8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

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