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

Question

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

Vedantu Content Team · Accepted Answer

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 $$x \geqslant 0$$ and the range is $$y \geqslant 0$$.We can graph $$y = \sqrt x $$ as follows:\n    \n    \n    \n    \n  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:\n    \n    \n    \n    \n  Now we will move the graph of $$y = \sqrt {x - 1} $$ by $$3$$ units down to obtain the graph of $$y = \sqrt {x - 1}  - 3$$.\n    \n    \n    \n    \n  Therefore, above graph is the required graph of $$y = \sqrt {x - 1}  - 3$$.Here, the domain of $$y = \sqrt {x - 1}  - 3$$ is $$x \geqslant 1$$ and range is $$y \geqslant  - 3$$.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.$$(6)$$ $$\dfrac{{f(x)}}{a}$$ 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.

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