We can write this equation in slope-intercept form
#y=mx+b#
where #m# is the slope and #b# is the #y#-intercept.
We know #m=-4# and #b=6# , so we can insert these into the equation to get
#y=-4x+6#
Hope this helps!
Video transcript
We are asked to graph y is equal to 1/3x minus 2. Now, whenever you see an equation in this form, this is called slope-intercept form. And the general way of writing it is y is equal to mx plus b, where m is the slope. And here in this case, m is equal to 1/3-- so let me write that down-- m is equal to 1/3, and b is the y-intercept. So in this case, b is equal to negative 2. And you know that b is the y-intercept, because we know that the y-intercept occurs when x is equal to 0. So if x is equal to 0 in either of these situations, this term just becomes 0 and y will be equal to b. So that's what we mean by b is the y-intercept. So whenever you look at an equation in this form, it's actually fairly straightforward to graph this line. b is the y-intercept. In this case it is negative 2, so that means that this line must intersect the y-axis at y is equal to negative 2, so it's this point right here. Negative 1, negative 2, this is the point 0, negative 2. If you don't believe me, there's nothing magical about this, try evaluating or try solving for y when x is equal to 0. When x is equal to 0, this term cancels out and you're just left with y is equal to negative 2. So that's the y-intercept right there. Now, this 1/3 tells us the slope of the line. How much do we change in y for any change in x? So this tells us that 1/3, so that right there, is the slope. So it tells us that 1/3 is equal to the change in y over the change in x. Or another way to think about it, if x changes by 3, then y would change by 1. So let me graph that. So we know that this point is on the graph, that's the y-intercept. The slope tells us that if x changes by 3-- so let me go 3 three to the right, 1, 2, 3-- that y will change by 1. So this must also be a point on the graph. And we could keep doing that. If x changes by 3, y changes by 1. If x goes down by 3, y will go down by 1. If x goes down by 6, y will go down by 2. It's that same ratio, so 1, 2, 3, 4, 5, 6, 1, 2. And you can see all of these points are on the line, and the line is the graph of this equation up here. So let me graph it. So it'll look something like that. And you're done.
The Algebra of Lines:
In this lesson, we learn how to graph our line using the y-intercept and the slope. First, we know that the y-intercept (b) is on the y-axis, so we graph that point. Next, we use the slope to find a second point in relation to that intercept. The following video will show you how this is done with two examples.
Video Source (05:37 mins) | Transcript
Steps for graphing an equation using the slope and y-intercept:
- Find the y-intercept = b of the equation y = mx + b.
- Plot the y-intercept. The point will be (0, b).
- Find the slope=m of the equation y = mx + b.
- Make a single step, using the rise and run from the slope. (Make sure you go up to the right if it’s positive and down to the right if it’s negative.)
- Connect those two points with your line.
Additional Resources
- Khan Academy: Intro to Slope-intercept Form (08:59 mins, Transcript)
- Khan Academy: Graph from Slope-intercept Equations (03:01 mins, Transcript)
- Khan Academy: Slope-intercept Examples (03:45 mins, Transcript)
Practice Problems
- Plot the line \({\text{y}}=-3{\text{x}}+2\) starting with the y-intercept and then using the slope.
- Plot the line \({\text{y}}=\frac{1}{2}{\text{x}}-3\) starting with the y-intercept and then using the slope.
- Plot the line \({\text{y}}=-\frac{3}{5}{\text{x}}+1\) starting with the y-intercept and then using the slope.
- Plot the line \({\text{y}}=2{\text{x}}+3\) starting with the y-intercept and then using the slope.
- Plot the line \({\text{y}}=-{\text{x}}-4\) starting with the y-intercept and then using the slope.
- Plot the line \({\text{y}}=\frac{4}{5}{\text{x}}+4\) starting with the y-intercept and then using the slope.
Graphing a Line using the Slope and y-intercept
To graph a line using its slope and y-intercept, we need to make sure that the equation of the line is in the Slope-Intercept Form,
From this format, we can easily read off both the values of the slope and y-intercept. The slope is just the coefficient of variable x which is m, while the y-intercept is the constant term b.
Here’s a quick diagram to emphasize this idea.
When these two pieces of information are identified, we are guaranteed to successfully graph the equation of the line.
- Plot the y-intercept \left( {0,b} \right) in the xy axis. Remember, this point always lies on the vertical axis y.
- Starting from the y-intercept, find another point using the slope. Slope contains the direction how you go from one point to another.
The numerator tells you how many steps to go up or down (rise) while the denominator tells you how many units to move left or right (run).
- Connect the two points generated by the y-intercept and the slope using a straight edge (ruler) to reveal the graph of the line.
Examples of Graphing a Line using the Slope and y-intercept
Example 1: Graph the line below using its slope and y-intercept.
Compare y = mx + b to the given equation \large{y = {3 \over 4}x - 2}. Clearly, we can identify both the slope and y-intercept. The y-intercept is simply b = - 2 or \left( {0,2} \right) while the slope is \large{m = {3 \over 4}}.
Since the slope is positive, we expect the line to be increasing when viewed from left to right.
- Step 1: Let’s plot the first point using the information given to us by the y-intercept which is the point \left( {0, - 2} \right).
- Step 2: From the y-intercept, find another point using the slope. The slope is m = {3 \over 4}, that means, we go up 3 units and move to the right 4 units.
- Step 3: Connect the two points to graph the line.
Example 2: Graph the line below using its slope and y-intercept.
I know that the slope is \large{m = {{ - 5} \over 3}} and the y-intercept is b = 3 or \left( {0,3} \right). Since the slope is negative, the final graph of the line should be decreasing when viewed from left to right.
- Step 1: Begin by plotting the y-intercept of the given equation which is \left( {0,3} \right).
- Step 2: Use the slope \large{m = {{ - 5} \over 3}} to find another point using the y-intercept as the reference. The slope tells us to go down 5 units and then move 3 units going to the right.
- Step 3: Draw a line passing through the points.
You might also be interested in:
Three Ways to Graph a Line
Graphing a Line using Table of Values
Graphing a Line Using X and Y
intercepts