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My professor just took up notes regarding curved lines and surfaces. In the curved lines section, he said a curved line on the 2D plane is an equation with x and y. He followed by saying if we replace x with x-a, the graph is shifted to the right by a. So, if we look at y = 2x.

x| 0 1 2 3 4
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y| 0 2 4 6 8

Let a = 1.

x-1| -1 0 1 2 3
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y  | -2 0 2 4 6

The graph simply moved down and left by 1 (a).

If we look at y = x^2.

x| 0 1 2 3
----------
y| 0 1 4 9

Let a = 1.

x-1| -1 0 1 2
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y  |  1 0 1 4

The graph just shrunk by 1 (a). What does he mean when he says "if we replace x with x-a, the graph is shifted to the right by a"?

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You appear to have misunderstood what your professor told you. Taking your first example of $y=2x$, we replace $x$ by $x-a$ in the equation to get the new equation $y=2(x-a)=2x-2a$. If you plot the resulting graph for $a=1$, you’ll see that it is indeed the original line shifted one unit to the right. Similarly, substituting into $y=x^2$ produces $y=(x-1)^2=x^2-2x+1$. If you plot this, you’ll see that it’s the original parabola shifted one unit to the right.

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This is not completely wrong. The remaining to recall how to plot a graph: to find out what $y$ is for each $x$.

For your first example, translating $y=2x$ to the right by replacing $x$ with $x-1$:

$$\begin{array}{c|c|c|c|c} x-1 & -1 & 0 & 1&2&3\\\hline y = 2(x-1) & -2&0&2&4&6\\\hline \color{red}x & 0 & 1&2&3&4 \end{array}$$

When you plot the $y$ values for each $\color{red}x$, the graph is indeed translated $1$ unit to the right. (e.g. the root is now $x=1$)

Another example, translating $y = x^2$ to the right by $1$ unit:

$$\begin{array}{c|c|c|c} x-1 & -1 & 0 & 1&2\\\hline y = (x-1)^2 & 1&0&1&4\\\hline \color{red}x & 0 & 1&2&3 \end{array}$$

Now try to plot the $xy$ graph.

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