Graphing a Line Using the Slope and Y-Intercept

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 \(y=−3x+2\) starting with the y-intercept and then using the slope. (
Solution

x

Solution:
Note: The graph you create may look slightly different depending on the spacing you choose for your x and y-axis. The correct graph should still have the same direction of slope and the x and y-intercepts should be the same.

Details:
To graph this line we need to identify the slope and the y-intercept. The equation is written in slope-intercept form, y=mx+b, where m is the slope and b is the y-intercept.

Step 1: Find the slope and the y-intercept of the line:

The equation of the line is

\({\text{y}}={\color{red}-3}{\text{x}}{\color{Blue}+2}\)

So the slope is \({\color{red}-3}\), and the y-intercept is \({\color{Blue}2}\).

Step 2: Graph the y-intercept:

This is a picture of a coordinate plane with the point (0,2) graphed on it.

Step 3: Find another point on the line using the slope:

The slope is \(−3\), which we can rewrite as \(-\dfrac{3}{1}\). Slope is \(\dfrac{\text{rise}}{\text{run}}\), which means that to find another point on the graph, we start at the y-intercept and then move down three spaces, then one space to the right:

This is a picture of a coordinate plane. The point \((0,2)\) has been graphed. There is an arrow pointing down three spaces from \((0,2)\) to the point \((0,−1)\). There is another arrow pointing from \((0,−1)\) to the point \((1,−1)\).

Step 4: Draw a line that passes through the points:

This is a picture of a coordinate plane with the points \((0,2)\) and \((1,−1)\) graphed on it. There is a line passing through both points.

)
Plot the line \(y=\dfrac{1}{2}x−3\) starting with the y-intercept and then using the slope. (
Solution

x

Solution:
Note: The graph you create may look slightly different depending on the spacing you choose for your x and y-axis. The correct graph should still have the same direction of slope and the x and y-intercepts should be the same.

)
Plot the line \(y=−\dfrac{3}{5}x+1\) starting with the y-intercept and then using the slope. (
Solution

x

Solution:
Note: The graph you create may look slightly different depending on the spacing you choose for your x and y-axis. The correct graph should still have the same direction of slope and the x and y-intercepts should be the same.

)
Plot the line \(y=2x+3\) starting with the y-intercept and then using the slope. (
Solution

x

Solution:
Note: Your graph may look a little different depending on the spacing you choose for your x and y-axis. Notice in this graph the hash marks for the x-axis are farther apart than the hash marks for the y-axis. This artificially makes the graph look less steep than it is if the hash marks are the same distance apart. However, sometimes this is helpful in order to better fit the data into the graph.

)
Plot the line \(y=−x−4\) starting with the y-intercept and then using the slope. (
Solution

x

Solution:
Note: In this graph, the spacing of the hash marks on the x and y-axis are spaced almost identically.

Details:
To graph this line we need to identify the slope and the y-intercept. The equation is written in slope-intercept form, \(y=mx+b\), where m is the slope and b is the y-intercept.

Step 1: Find the slope and the y-intercept of the line:

The equation of the line is

\({\text{y}}={{\color{Red} -\text{x}}}{\color{Blue}{-}4}\)

Keep in mind that \({\color{red}-}{\color{Red}{\text{x}}}\) is equal to \({\color{red}-1}{\color{Red}{\text{x}}}\), so an equivalent equation is:

\({\text{y}}={\color{red}-1}{\text{x}}{\color{Blue}{-}4}\)

So the slope is \({\color{red}-1}\), and the y-intercept is \({\color{blue}-4}\)

Step 2: Graph the y-intercept:

This is a picture of a coordinate plane with the point \((0,−4)\) graphed on it.

Step 3: Find another point on the line using the slope:

The slope is \(−1\), which we can rewrite as \(-\dfrac{1}{1}\). Slope is \(\dfrac{\text{rise}}{\text{run}}\), which means that to find another point on the graph, we start at the y-intercept and then move down one space, then one space to the right:

This is a picture of a coordinate plane with the points \((0,−4)\) and \((1,−5)\) graphed on it. There is an arrow pointing from \((−4,0)\) to \((−5,0)\) and an arrow pointing from \((−5,0)\) to \((1,-5)\)

Step 4: Draw a line that passes through the points:

This is a picture of a coordinate plane with the points \((0,−4)\) and \((1,−5)\) graphed on it. There is a line passing through both points.

)
Plot the line \(y=\dfrac{4}{5}x+4\) starting with the y-intercept and then using the slope. (
Solution

x

Solution:

)

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