Question

How do you graph `y=(1/5)^x` and `y=(1/5)^(x-2)` and how do the graphs compare?

Answer 1

Graph them by manually calculating points and plotting them, or use a spreadsheet or plotting program.

Explanation:

No matter where you put the range of values, the relative shape of the two curves remains the same. Both are inverse exponential curves (logarithmic curves) with an asymptote at y = 0.
`y = (1/5)^x and y = (1/5)^(x−2)`

Answer 2

See below.

Explanation:

Graph 1:

`y=(1/5)^x`

`y = 5^-x`

This has the graph of the standard negative exponential function `f(x) =a^-x`; where `a=5`. The properties of such a graph are as follows:

• The graph passes through the point `(0,1)`
• The domain is `(-oo,+oo)`
• The range is `(0, +oo)`
• The graph is decreasing
• The graph is asymptotic to the x-axis as `x -> +oo`
• The graph increases without bound as `x -> -oo`
• The graph is smooth and continuous.

This graph is shown below.

graph{(1/5)^x [-10, 10, -5, 5]}

Graph 2:

`y=(1/5)^(x-2)`

`y = 5^(-(x-2)) = 5^-x xx 5^2`

This is the Graph 1 above scaled by 25 as shown below..

graph{y=(1/5)^(x-2) [-10, 10, -5, 5]}

This graph has all the properties of Graph 1 above except that it passes through the point `(0,25)`