A question on shifting and scaling graphs (in calculus, algebra) I am reviewing high school algebra before studying the university calculus.
I have a question about shifting and scaling graphs.
My problem is as follows.
$$y=\sqrt{1-{x\over2}}$$
I need to sketch the graph of this function when I already know the graph of $y=\sqrt x$.
First, I rewrote the expression.$$y=\sqrt{{-1\over2}(x-2)}$$
Incorrect Answer
I thought I needed to shift the graph by 2 units, and then stretch the graph in the $x$ direction by 2. Then I tried to reflect the graph across the $y$-axis.
Correct Answer
Actually, I know the correct answer. I need to reflect the graph across the $y$-axis first. Then I need to stretch the graph in the $x$ direction by 2. Finally, I need to shift the graph by 2 units to the right.
The problem is, I don't understand why I should do it this way. Why is the first approach a wrong answer? Please help me figure out this problem.
 A: 
$$y=\sqrt{1-{x\over2}}$$


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Correct Answer I don't understand why I should do it this way.
I need to reflect the graph $y=\sqrt x$ across the $y$-axis first.

Start with the graph of $$y=f(x).$$ Applying the above reflection
gives $$y=f(-x).$$

Then I need to stretch the graph in the $x$ direction by 2.

This then gives $$y=f\left(-\left(\frac12x\right)\right).$$

Finally, I need to shift the graph by 2 units to the right.

This finally gives
$$y=f\left(-\left(\frac12(x-2)\right)\right)\\=f\left(1-\frac
x2\right),$$ as required.


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Incorrect Answer Why is this approach wrong?
I thought I needed to shift the $y=\sqrt x$ graph by 2 units, and then stretch the graph in the $x$ direction by 2. Then I tried to
reflect the graph across the $y$-axis.

Starting again with the graph of $$y=f(x),$$ the above sequence of
transformations gives $$y=f(x-2),$$ then
$$y=f\left(\left(\frac12x\right)-2\right),$$ then
$$y=f\left(\left(\frac12(-x)\right)-2\right)\\=f\left(-2-\frac
x2\right).$$
Here's a related discussion: How does one explain that transformations 'inside' a function operate in the opposite direction than intuition suggests?.
