So I have to sketch a graph of the function:
y= -2 log₁₀ x
Now I can write the log₁₀ x in exponential form as 10^y=x
But what do I do about the -2? Do I then do a table with x and y etc.?
Hi synchro,
To sketch a graph of any function you need to consider;
1. The value of $\displaystyle y$ when $\displaystyle x~=~0$ In this case undefined.
2. The value of $\displaystyle x$ when $\displaystyle y~=~0$ In this case $\displaystyle 1$
3. What happens to the value of $\displaystyle y$ as $\displaystyle x\rightarrow 0$ Note that the $\displaystyle \log$ of a fraction is negative, thus in this case $\displaystyle y\rightarrow \infty$
4. What happens to the value of $\displaystyle y$ as $\displaystyle x\rightarrow \infty $ In this case $\displaystyle y\rightarrow -\infty$
5. And the result for $\displaystyle x\rightarrow -\infty$ In this case undefined since the $\displaystyle \log$ of a negative number is undfined.
Constructing a table like that implies that your not actually sketching a graph, more drawing one.
The table should consist of 2/3 columns, namely $\displaystyle x$ , $\displaystyle \log_{10}x$ , $\displaystyle y$ of which the middle one can be erased according to your preference.
If you have the $\displaystyle x$ values then you can sub into the equation $\displaystyle y~=~-2\log_{10}x$ to find the $\displaystyle y$ values.
If you have the $\displaystyle y$ values then you can rearrange the equation accordingly to help you find the $\displaystyle x$ values;
We have $\displaystyle y~=~-2\log_{10}x $ Multiplying both sides by $\displaystyle -\frac{1}{2}$ gives $\displaystyle \log_{10}x~=~-\frac{1}{2} \cdot y~$ Finally raising both sides to a power of $\displaystyle 10$ (and a bit of rearranging) leads us to obtain the following equation $\displaystyle x~=~\frac{1}{\sqrt{10^y}}~$ Now subbing the $\displaystyle y$ values into this equation will give you the corresponding $\displaystyle x$ values.
Ahh okay, I was mistaken. Yep the stipulation is to "sketch a graph of". Could you please explain the "key points" in your first post in this thread? As in, why are they to be considered? How do you sketch a graph after having gone through that 'checking'? I don't really know much about sketching functions, of this type or otherwise (though I only need to know how to sketch of this type, i.e. with a "log" in).
Many thanks!!
to graph logs, the best way is to use transformations. that is, you should know what the general shape of a log graph is, and then you can shift, flip or stretch it to get the graph you want, provided the function is simple enough, which in this case, it is. see if this helps. you can search for similar threads, there are a lot