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How do you graph $y=-3\sqrt{x}-3$ compare to the parent graph, and state the domain and range?

Last updated date: 13th Jun 2024
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Hint: Here we have to graph $y=-3\sqrt{x}-3$ graph it by using a graphing calculator, if we do not have the graphing calculator, we can also use the Desmos calculator which we can use online and is free on the internet. There are buttons present at the bottom of the screen that can be used for graphing the function. We have the parent functions $\sqrt{x}$ that you can also graph to see how it compares to the transformed functions. We will figure out the domain and range graphically.

Complete step by step solution:
Here we have,
We have to solve it by graphing calculator, if do not have any then use desmos calculator,
Compare it to equation,
The parent graph or functions $\sqrt{x}$ can also be graphed only to see how it compares to your transformed functions.
The standard form for writing the function is
$a.\sqrt{b\left( x+c \right)}+d$
Whereas $a$ and $d$ are vertical transformers and $b$ and $c$ are horizontal transformations.
The main thing about this transformation is that it is reflected over $x$-axis. That can be seen in your function, because the leading coefficient $(a)$ is negative.
This graph is translated down to $3$ units. Which is shown in given equation because $d$ is negative $3.$
Then graph is vertically stretched by a factor of $3.$
Since, $\left| a \right|=3$
Now, for domain and range.
This can be figured out graphically, by looking at the graph and also seeing that $x$ is greater than $0$ and $y$ and also must be less than $-3$
So, by using graphing calculator, the domain is $\left\{ x|x\ge 0,x\in R \right\}$
and range is $\left\{ y|y\le -3,y\in R \right\}$
In this way we can graph $y=-3\sqrt{x}-3$ compared to the parent graph, and also find the domain and range.

Additional Information:
The parent graph is the graph of a relatively simple function by transforming the function in various ways. The graph can be translated or reflected.
The domain is a set of possible input values, the domain of graphs considering all the input values on the $x$-axis. In other cases, The range is the set for all the output values. The range of graphs showed all the output values on the $y$-axis. The domain and range is always written from smaller to larger values, from left to right for domain, and bottom to top is for range of graphs.

Solve the problem by graphing calculator if you do not have then download desmos calculator, after solving check the graph properly sometimes positive of domain and range is changed due to some mistake. Because the domain is from left to right and the range is from bottom to top. These are some key points which we have to remember.