3.06 Identifying the y-intercept
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).
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?
| x | y |
|---|---|
| -2 | 4 |
| -1 | 7 |
| 0 | 10 |
| 1 | 13 |
Examples
Example 1
Consider the following table of values.
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | -5 | -2 | 1 | 4 |
Identify the coordinates of the y-intercept.
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.
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.
Example 3
Consider the following graph of a lne:
What is the slope of the line shown in the graph?
What is the y- value of the y-intercept of the line shown in the graph?
Write the equation of the line in slope-intercept form.
The general slope-intercept form of a line: