I do not understand functions. I need help with the following equation. $f(x)=\sqrt{x+2} - 3$
a) What is the shape of the graph of this function?
b) How do we translate the graph of $\sqrt x$ to obtain $f(x)$
c) Sketch a graph of $f(x)$. Fill in values for the table, including the critical point of the graph (You must figure out the correct $x$ value).
- I have a table with $x$ and $f(x)$. I am given $x$ for $-3$ and want me to try different values for $x$ and derive $f(x)$ in my table. I need to pick 3 random numbers and input it.
Furthermore I also have to find the domain and the range of this graph.
To determine the domain and range of the function, you can use that the argument of the square root should always be greater than or equal to zero. So specifically, in this case, the only restriction you have on $x$ is that $x + 2 \geq 0$. The domain of the function consists of all $x \in \mathbb{R}$ for which this holds. Furthermore, for the range, you need to determine all the different values that $f(x)$ can take. For this, you can use that the square root of a real number is always greater than or equal to zero; $\sqrt{x+2} \geq 0$ for all $x$ in the domain of $f$. This gives you a lower bound for $f$: $f$ can take all the values greater than this lower bound.
To see how you can get the graph of $f$ from the graph of $\sqrt{x}$, you first want to change this from $\sqrt{x} \rightarrow \sqrt{x+2}$; this is a horizontal translation. For this, you need to shift the original graph 2 units to the left. (Can you see why?) The $-3$ is a vertical translation; can you think of how you need to shift the graph of $\sqrt{x+2}$ to get the graph of $\sqrt{x+2} -3$? (You can always plot your graphs on http://www.wolframalpha.com/ to check if what you are doing is correct. However, it is of course important to think about it and understand why it is or isn't correct.)
Finally, for a sketch of the graph, first try to fill in your table and compute $f(x)$ for some values of $x$. You should then be able to draw the graph of the function.