Quadratic Functions - Graph

[kduvernay]

I am doing a practice SAT problem that I cannot figure out. I have seen this type of problem a few times and cannot seem to find how to get the solution. The problem will show a graph that is irregular (not a line, parabola) and state that is the graph of y=f(x). Then it will ask which of the following graphs is y=f(x+1) or y=2f(x) and show a bunch of graphs that are similar to the original, but shifted up, down, right, left, or stretched. I know from looking up the answer that the graph of y=f(x+1) shifts the graph of y=f(x) 1 unit to the left. However, I don't understand why. Any insight on this type of problem would be greatly appreciated!

 

[mmm4444bot]

… I know from looking up the answer that the graph of y=f(x+1) shifts the graph of y=f(x) 1 unit to the left. However, I don't understand why …

Something is wrong, in so far as you had to look up the answer in order to learn this fact about shifting graphs. I mean, somebody is responsible for teaching you this before assigning such an exercise.

Here's one way to think about it.

f(x + 1) gives the behavior of f(x) at x + 1, but that behavior is plotted at x.

Move from left to right along the x-axis, and stop somewhere. Now, look ahead one unit "to the future" to see what f(x) is doing there. Plot what you see right where you're at.

Now move along the x-axis a little.

Again, look forward "in the future" one unit to see what f(x) is doing one unit in front of you. Plot that behavior right where you're at.

Move ahead a little more.

Look forward one unit, and plot that behavior right where you're at.

Move ahead again, and so on, and so on.

As you move from left to right, you are always looking at f(x) one unit in front of you to see what's coming up, and then you're "pulling" this behavior back toward yourself one unit to plot.

Since this happens at every x, the entire behavior of f(x) is being "pulled" (shifted) one unit to the left.

f(x + 1) gives the behavior of f(x) at x + 1, but that behavior is plotted one unit to the left at x.

(The situation is somewhat cerebral because it's abstract.)