# How to account for stretching in graph transformation of $y = \sqrt{x}$?

Hi guys, I got this question as an assignment from my university and have been trying to solve it for a whole day now but can't get the correct answer. Well, I know how to shift Graph towards left, right, upwards and downwards $C$ points. Well for example current Graph's equation is $y = x^2$ and if we have to move graph to left $C$ points. We will add $C$ to the function and it will become $y = (x+C)^2$ and if we have to shift Graph to right side $C$ points, then we will subtract $C$ from function, and the equation will become $y = (x-C)^2$, and similarly add $C$ to the $f(x)$ and subtract $C$ from $f(x)$ if we want to shift graph upwards $C$ points and downwards $C$ points respectively, and if we want to reflect it about $x$-axis, we will multiply $f(x)$ with minus "-" sign and if we want to reflect it about $y$-axis we will multiply the function with minus"-" sign thats makes it like $y = f(-x)$. Well now you now that I know these things. But I have been trying to solve this question and its been a day but I am not able to solve it correctly. Kindly help me, I want to submit my Assignment and I really do not want to lose even a single marks from Mathematics.

And yeah I forgot to tell that I have already figured out that the graph in second picture is reflected about $x$-axis and then moved $1$ unit/point toward left side and moved $1$ unit/point downwards, but if you look carefully the graph in second picture is little stretched as well and thats where I am getting problem because I am not figuring out that is it stretched towards x-direction or y-direction and also can't figure out that how many points/units it have stretched.

The original graph has been stretched by factor $3$ in the $y$ direction, reflected in the $x$ axis, and translated by $-1$ in both $x$ and $y$ directions.

Therefore your new graph is $$y=-1-3\sqrt{x+1}$$

Good work so far.

I am not figuring out that is it stretched towards x-direction or y-direction

Here's how to get started on that.

The transformations that stretch are the ones that replace $x$ by $ax$ or $y$ by $by$. To figure out which stretch you need and the value of the constant, look at a few easy interesting points on the graphs (after you've reflected and translated so that the new one starts at the origin and heads off into the first quadrant).

Hint: We know that $f(x) = \sqrt{x}$ grows from $0$ to $1$ as $x$ increases from $0$ to $1$ and that $f(x) = \sqrt{x}$ grows from $0$ to $2$ as $x$ increases from $0$ to $4$. If we call the second function $g$, then $g$ decreases by $3$ as $x$ increases from $-1$ to $0$ (one unit to the right of the endpoint) and decreases by $6$ as $x$ increases from $-1$ to $3$ (four units to the right of the endpoint). What factor do you need to multiply the square root by in order to account for this transformation?