How Do You Graph An Absolute Value Function in Desmos and WolframAlpha?

Graphing an absolute value function can be a bit deceiving. Depending on the technology you use, the graph you get may not actually represent the function well.

Let’s graph the function

using WolframAlpha and Desmos. These two online graphing tools are both free to use and can produce excellent graphs.

To graph this function in WolframAlpha, go to the website and type this in the box on the screen.

Both the numerator and denominator need to be in parentheses. The absolute value function in WolframAlpha is “abs”. Putting this in front of (x+4) means the absolute value of the quantity x + 4.

Press return to give the following result.

The graph consists of a horizontal section at y = -1 and another at y = 1. These sections are connected by a vertical line at x = -4. This is problematic since this is not a function…it does not pass the vertical line test at x = -4.

Let’s try graphing this function in Desmos.

As shown in the video above, the graph of this function looks like this in Desmos.

This looks similar to the WolframAlpha version, except that the tow horizontal pieces are not connected. As noted in the video,

is undefined at x = -4. This is because x = -4 causes the denominator to be zero.

This might not seem like a big deal. But if you were determining whether the function was continuous at x = -4, the two graphs would lead to different conclusions. The WolframAlpha graph would lead you to think the function is continuous. Desmos would give the opposite conclusion.

In this case, Desmos gives a more accurate graph since it shows the discontinuity at x = -4. An even better version of this graph would be to include open circles at x = -4.

This not only shows the discontinuity, but also indicates that the function is undefined at x = -4. To put these on the graph I downloaded the image and then added the circles in an image editing program like Paint.

How Do You Evaluate The Limit Of A Difference Quotient?


Problem Evaluate the difference quotient  for f (x) = x2 – 2x + 4.

This is a little different from   but works the same way. Since a value is not supplied for x, we just leave it and work out the limit. Start by evaluating  f (x + h):

Make sure you FOIL the square out and distribute the negative.

Now put this along with f (x)  into the difference quotient.

As h gets smaller and smaller, the term in the middle gets smaller. This means the limit is equal to 2x – 2. Since the other terms do not contain x, they are unaffected when h gets small.

How Do You Make a Table of Values in Google Sheets?

Suppose you want to generate a table of values from a formula…perhaps to help evaluate a limit. Google Sheets (or any spreadsheet) can quickly generate the values.

Make a table of values for f (x) = 7x2 + 1 for x = 0, 0.9, 0.99, 0.999, 0.9999.

1.  Open Google Sheets.

2.  In the first row place the name of the input variable and the function’s name. We will put the values of the independent variable the first column (A) and the values of the dependent variable in the second column (B).


3.  In cells A2 through A6, put the values of the independent variable for the table.


4.  Click the mouse in cell B2. Enter the formula in that cell by typing =7*A2^2+1.


5.  Press Enter to evaluate the formula in cell B2.


6.  Click the mouse in cell B2. You will see a black outline around the cell. Use the mouse to grab the fill handle in the lower right hand corner of the black outline.


7.  Click on this handle. While holding the left mouse button down, drag the mouse to cell B6.


8.  When the mouse button is released, cells B2 through B6 are filled with the function values corresponding to the inputs in column A.