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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.

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  • $\begingroup$ Break it down step by step. Correct: $\,\sqrt{x} \;\mapsto\; \sqrt{-x} \;\mapsto\; \sqrt{-x/2} \;\mapsto\; \sqrt{-(x-2)/2}\,$. Incorrect: $\dots$ $\endgroup$
    – dxiv
    Jan 11 at 22:44
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$$y=\sqrt{1-{x\over2}}$$

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

Here's a related discussion: How does one explain that transformations 'inside' a function operate in the opposite direction than intuition suggests?.

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