y=a/x, x≠0
Let's see what happens when x takes negative values.
x
a/x
y=a/x
- 100
a/- 100
- 0.01a
- 10
a/- 10
- 0.1a
- 1
a/- 1
- a
- 0.1
a/- 0.1
- 10a
- 0.001
a/- 0.001
- 100a
Recall that, in this case, we have that a>0. Therefore, as the x-variable increases, the value of y decreases. Let's see now what happens when x takes positive values.
x
a/x
y=a/x
0.001
a/0.001
100a
0.1
a/0.1
10a
1
a/1
a
10
a/10
0.1a
100
a/100
0.01a
In the above table, we can also see that y decreases as x increases. Therefore, if a>0, the graph of the function y= ax decreases within both halves of its domain. Considering that the asymptotes are x=0 and y=0, we can graph the function.
b We will use a similar approach as in Part A. Since in this case a is a negative number, it can be written as a=- b, with b>0.
y=a/x ⇔ y=- b/x, x≠0
Let's make a table for negative values of x.
x
- b/x
y=- b/x
- 100
- b/- 100
0.01b
- 10
- b/- 10
0.1b
- 1
- b/- 1
b
- 0.1
- b/- 0.1
10b
- 0.001
- b/- 0.001
100b
Since b>0, we can see that as the x-variable increases, the values of y also increase. Finally, let's see what happens when x takes positive values.
x
- b/x
y=- b/x
0.001
- b/0.001
- 100b
0.1
- b/0.1
- 10b
1
- b/1
- b
10
- b/10
- 0.1b
100
- b/100
- 0.01b
We see that as x increases, y also increases. Therefore, the graph of y= - bx increases in both halves of its domain.
y=- b/x ⇔ y=a/x, x≠0
We conclude that, for a<0, the graph of y= ax increases in both halves of its domain. Again, considering that the asymptotes are x=0 and y=0, we can graph the function.
