Similarities: Each graph is composed of two curves with asymptotes at x=- 2 and y=0. Differences: The graph of y= - 3x+2 is a reflection across the x-axis of the graph of y= 3x+2.
Practice makes perfect
We are asked to describe the similarities and differences between the graphs of two rational functions.
y= 3x+2 and y= - 3x+2
To do so, we will first draw the graphs of the functions and then compare them. Let's do it!
Graphing the Functions
Here are the steps we can follow to graph a rational function.
Make a table using values from both sides of the vertical asymptote.
Draw the asymptotes and plot the points from the table on the same coordinate plane. Connect the points on either side of the vertical asymptote with smooth curves.
We will complete these steps for both our functions simultaneously.
Identifying the Asymptotes
Note that the given functions can be written in the following form.
y=a/x-b+ c
The graph of a rational function in this form has a vertical asymptote at x=b and a horizontal asymptote at y= c. Now let's identify the values of b and c for both our functions. This will let us determine the asymptotes of the functions.
y=3/x+2
y=- 3/x+2
Rewrite
y=3/x-(- 2)+0
y=- 3/x-(- 2)+0
b
- 2
- 2
c
0
0
We can see that both functions have the same values of b and c. Therefore, both functions have a vertical asymptote at x=- 2 and a horizontal asymptote at y= 0, which is the x-axis.
Making the Table
Since the functions have the same vertical asymptote, we can use the same x-values to find the corresponding y-values for each function. For both functions, let's find the coordinates of three points on the left-hand side of the vertical asymptote and of three points on its right-hand side.
x
3/x+2
- 3/x+2
- 5
3/- 5+2=- 1
- 3/- 5+2=1
- 4
3/- 4+2=- 1.5
- 3/- 4+2=1.5
- 3
3/- 3+2=- 3
- 3/- 3+2=3
- 1
3/- 1+2=3
- 3/- 1+2=- 3
0
3/0+2=1.5
- 3/0+2=- 1.5
1
3/1+2=1
- 3/1+2=- 1
Drawing the Graphs
Finally, we will draw the graphs of our functions. Let's start by drawing the asymptotes: x=- 2 and y= 0.
Now we will plot the points from the table.
Last, let's connect the points to form the two graphs.
Comparing the Graphs
As we mentioned previously, the given functions share the same asymptotes.
Vertical Asymptotes:& x=- 2
Horizontal Asymptotes:& y= 0
Also, each graph is composed of two curves.
We can notice that the graph of y= -3x+2 is a reflection across the x-axis of the graph of y= 3x+2.
This can be confirmed algebraically by putting the negative from the numerator in front of the fraction.
y=- 3/x+2 ⇔ y=-3/x+2
