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How do I sketch the following functions, applying appropriate transformations to the graphs of $y= \frac{1}{x} ,y= \sqrt x ,y=x^2$.

$$f(x)=\frac{x+1}{x}=1+\frac{1}{x}; g(x)=\sqrt{x-4};h(x)=(x+1)^2 -3.$$

I am absolutely lost on this question and don't know where to start. Any help is very much appreciated :).

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  • $\begingroup$ There is a typo in your function $f$. $\endgroup$ – user587192 Aug 29 '18 at 0:47
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    $\begingroup$ A good way to get started may be looking at some examples. Have you seen any similar problems before from your notes/book? $\endgroup$ – user587192 Aug 29 '18 at 0:48
  • $\begingroup$ I would say that $g$ and $h$ are the easiest, so maybe start with them (maybe $h$ being the easiest). You should try these and show your attempts at a solution and highlight where you are encountering specific problems. Hint: for $h$, can you graph $y=x^2$? $\endgroup$ – Dave Aug 29 '18 at 0:52
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    $\begingroup$ And indeed, $f(x)=\color{red}{1}+\frac{1}{x}$ for $x\neq 0$. $\endgroup$ – Dave Aug 29 '18 at 0:54
  • $\begingroup$ Thank you for notifying my of that typo. I'll fix it and I don't understand what it is meant when it says 'applying appropriate transformations to the graphs of ...'. Is it just referring to the functions? $\endgroup$ – WBYT Aug 29 '18 at 1:05
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The task is to look at the graphs of the functions $y=\frac{1}{x}$, $y=\sqrt{x}$ and $y=x^2$ and then modify them.

For example if we take $y=\sqrt{x}$. You can easily sketch the graph and now if you consider the function $g$, you just need to shift the graph by $4$ units to the right.

The same with other functions. The idea is just to start with the basic function and then sketch the graph of $f, g, h$ by applying some basic transformations (in the example - substracting a constant from the argument).

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1.for $f(x)$ first sketch the graph of $\dfrac{1}{x}$ then shift it 1 unit upward. graph of $\dfrac{1}{x}$ graph of $1+\dfrac{1}{x}$

2.for $g(x)$ first sketch the graph of $\sqrt{x}$, then shift it 4 unit to the right. graph of $\sqrt{x}$ graph of $\sqrt{x-4}

3.for $h(x)$ first sketch the graph of $x^2$ then shift it 1 unit to the left and then shift it 3 unit downward. graph of $x^2$ graph of $(x+1)^2$ graph of $(x+1)^2-3$

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