Graphing translations

alexnixon · **Joined:** Thu, 8 Sep 2011 20:16:42 UTC **Posts:** 1

I need help graphing translations of logrithmic functions and exponential functions. things like f(x) = log1/5 x where 1/5 is the base and f(x)=(log5 x) +2 where 5 is the base. I just don't remember the rules of translation, like what all the negatives and things mean. and same for exponential functions like f(x)= (1/2)^-x and f(x)= (1/1/2)^x. any help would be good.

thanks

Shadow · **Posted:** Thu, 8 Sep 2011 21:46:45 UTC

alexnixon wrote:

I need help graphing translations of logrithmic functions and exponential functions. things like f(x) = log1/5 x where 1/5 is the base and f(x)=(log5 x) +2 where 5 is the base. I just don't remember the rules of translation, like what all the negatives and things mean. and same for exponential functions like f(x)= (1/2)^-x and f(x)= (1/1/2)^x. any help would be good.

thanks

Recall $\left({a\over b}\right^x={a^x\over b^x}$ and $a^{-x}={1\over a^x}$ .

Also $\log(ax)=\log a+\log x$

Soroban · **Posted:** Fri, 9 Sep 2011 03:41:11 UTC

Hello, alexnixon!

Quote:

I need help graphing translations of logrithmic functions and exponential functions.

$(1)\;f(x) \:=\: \log_{\frac{1}{5}}x$

$(2)\;f(x)\:=\:\log_5x +2$

I assume you know the graph of $f(x) \:=\:\log_5x$
. . Call it Graph 1.

From (1), we have: . $y \:=\:\log_{\frac{1}{5}}x$

Then: . $\left(\frac{1}{5}\right)^y \:=\:x \quad\Rightarrow\quad \left(5^{-1}\right)^y \:=\:x \quad\Rightarrow\quad 5^{-y} \:=\:x$

$\text{Take logs, base 5: }\:\log_5\left(5^{-y}\right) \:=\:\log_5x \quad\Rightarrow\quad -y\underbrace{\log_55}_{\text{This is 1}} \:=\:\log_5x$

We have: . $-y \:=\:\log_5x \quad\Rightarrow\quad y \:=\:-\log_5x$

This is the negative of Graph 1, reflected over the x-axis.

(2) is Graph 1 moved up 2 units.

Soroban · **Posted:** Fri, 9 Sep 2011 03:48:30 UTC

Hello again, alexnixon!

Quote:

I need help graphing translations of exponential functions.

$(3)\;f(x)\:=\: \left(\frac{1}{2}\right)^{-x}$

$(4)\;f(x) \:=\: \left(\frac{1}{\frac{1}{2}}\right)^x$

I assume you know the graph of $f(x) \:=\:2^x$
. . Call it Graph 2.

From (3) we have: . $f(x) \:=\:\left(\frac{1}{2}\right)^{-x} \:=\:\left(2^{-1}\right)^{-x} \:=\:2^x$

The graph of (3) is identical to Graph 2.

From (4) we have: . $f(x) \:=\:\left(\frac{1}{\frac{1}{2}}\right)^x \:=\:2^x$

The graph of (4) is identical to Graph 2.

S.O.S. Mathematics CyberBoard

Graphing translations

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