# 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.

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.

• Break it down step by step. Correct: $\,\sqrt{x} \;\mapsto\; \sqrt{-x} \;\mapsto\; \sqrt{-x/2} \;\mapsto\; \sqrt{-(x-2)/2}\,$. Incorrect: $\dots$
– dxiv
Jan 11 at 22:44

$$y=\sqrt{1-{x\over2}}$$

1. 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.

2. 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 $$y=f\left(\left(\frac12(-x)\right)-2\right)\\=f\left(-2-\frac x2\right).$$