Question

How do you graph $$y=\sqrt{x-1}-3$$?

Answer 1

Hint: To graph $$y=\sqrt{x-1}-3$$, we know that the parent function of the functions of the form $$f(x)=\sqrt{x-a}+b$$ is $$f(x)=\sqrt{x}$$. So, we will start with the graph of $$y=\sqrt{x}$$. Then change $$x$$ to $$x-1$$ to obtain $$y=\sqrt{x-1}$$. Then we will translate the graph $$3$$ units down to obtain the graph of $$y=\sqrt{x-1}-3$$.

Complete step by step answer:
We have to graph $$y=\sqrt{x-1}-3$$. We can also write this as $$y-\left(-3\right)=\sqrt{x-1}$$.
The parent function of the functions of the form $$f(x)=\sqrt{x-a}+b$$ is $$f(x)=\sqrt{x}$$.
Here, the domain of $$f(x)=\sqrt{x}$$ is $$x⩾0$$ and the range is $$y⩾0$$.
We can graph $$y=\sqrt{x}$$ as follows:

Now we will move the graph of $$y=\sqrt{x}$$ by $$1$$ unit to the right to obtain the graph of $$y=\sqrt{x-1}$$.
So, the graph of $$y=\sqrt{x-1}$$ is given as follows:

Now we will move the graph of $$y=\sqrt{x-1}$$ by $$3$$ units down to obtain the graph of $$y=\sqrt{x-1}-3$$.

Therefore, above graph is the required graph of $$y=\sqrt{x-1}-3$$.
Here, the domain of $$y=\sqrt{x-1}-3$$ is $$x⩾1$$ and range is $$y⩾-3$$.

Note:
Some rules to transform the given graph of a function that we should keep in mind are as follows:
$$(1)$$ $$f(x+a)$$ horizontally shifts the graph of $$f(x)$$ to the left by $$a$$ units.
$$(2)$$ $$f(x-a)$$ horizontally shifts the graph of $$f(x)$$ to right by $$a$$ units.
$$(3)$$ $$f(x)+a$$ vertically shifts the graph of $$f(x)$$ upward by $$a$$ units.
$$(4)$$ $$f(x)-a$$ vertically shifts the graph of $$f(x)$$ downward by $$a$$ units.
$$(5)$$ $$af(x)$$ vertically stretches the graph of $$f(x)$$ by a factor of $$a$$ units.
$$(6)$$ $${\displaystyle \frac{f(x)}{a}}$$ vertically shrinks the graph of $$f(x)$$ by a factor of $$a$$ units.
$$(7)$$ $$f(ax)$$ horizontally shrinks the graph of $$f(x)$$ by a factor of $$a$$ units.
$$(8)$$ $$f(ax)$$ horizontally stretches the graph of $$f(x)$$ by a factor of $$a$$ units.