Question

Write an equation for the given graph. The graph represents logarithmic function f with base 2 or 3, translated and or reflected.

Answer 1

The graph represent the following function translated or reflected:

`f(x)=\log _2x\text{ or }\log _3x`

Lets calculat some values for f(x), considering base 2 or 3:

`\begin{gathered} \text{Base 2:} \\ f(1)=0 \\ f(2)=1 \\ f(4)=2 \\ \text{Base 3:} \\ f(1)=0 \\ f(3)=1 \\ f(9)=2 \end{gathered}`

We can check that both possible functions are defined only at x > 0, and tends to - infinity when x tends to 0

We also can check that the logarithm with base 3 grows slower than the logarithm with base 2

Now, if we take a look at the graph, we will check that there is a vertical asymptote at x = 3 and the graph of function is turned to the left side, and not to the right side, how the graph of a logarithmic function would look like.

Then. the graph represent the function f reflected on the y-axis and translated three units right

We also can check that when x = 2 (which would be equivalent to x = 1 if the function where not reflected or translated), we have y = -1, and not y = 0. Therefore, the function also is translated one unit down.

Whe also can conclude that the graph is related to a logarithmic function with base 2, since when x = 1 (which would be equivalent to x = 2 if the function were not reflected or translated), we have y = 0 (which would be y = 1 if it weren't translated one unit down).

Therefore, the equation for the function represented in the graph must be:

`f(-x+3)-1=\log _2(-x+3)-1_{}`