Create a graph that represents the function f (x − 3), if f (x) = x^2

@Mertsj

Question

Create a graph that represents the function f (x − 3), if f (x) = x^2

@Mertsj

whpalmer4 · Answer

Do you know what the graph of $y = f(x) = x^2$ looks like?

whpalmer4 · Answer

This will be a translation of that graph in one of four directions: up, down, right, or left.

whpalmer4 · Answer

Which one?

If we add a number to the result of $f(x)$, such as $y = f(x)+1$, we make each point appear 1 unit higher on the graph, right?  

If we subtract a number from the result of $f(x)$, such as $y = f(x)-1$, we make each point appear 1 unit lower on the graph.  Doesn't matter what the definition of $f(x)$ actually is; we shift the graph up or down by the amount added or subtracted from the output of $f(x)$.

anonymous · Answer

Uhh the second one?

whpalmer4 · Answer

You could think of it as plotting the graph of $y = f(x)$ on a sheet of clear plastic, and sliding it up and down over your graph paper.

Now, that's the easier pair of translations to understand.  Let's make a little table:

x     x^2    x^2+1
0        0       1
1         1       2
2         4       5
3        9         10
4        16       17

if you plot those points on a graph, drawing a separate curve through the x^2+1 points, you'll see that adding 1 to x^2 just reproduces the same curve, shifted up 1 unit.

whpalmer4 · Answer

Blue line is x^2, purple line is x^2+1

anonymous · Answer

Oh wait, it gives me options.. Should I place the options?

whpalmer4 · Answer

Now let's tackle the other case:  we add or subtract to/from x before squaring it

x       x+1      x^2       (x+1)^2
0         1          0             1
1         2          1              4
2          3         4             9
3          4         9             16
4          5         16           25

Do you see what is happening?  the (x+1)^2 column has the same values as the x^2 column, except where the x^2 column has a given value at x, the (x+1)^2 column has it at x-1!

whpalmer4 · Answer

Nah, we don't need no steekin' options, we're going to learn how to do this all by ourselves!

anonymous · Answer

First one A, second one B, third one C, fourth one D.

whpalmer4 · Answer

Here's a plot of those two curves:
Again, blue is x^2, purple is (x+1)^2

anonymous · Answer

Ok.

whpalmer4 · Answer

So when we add a number to the argument of the function, we shift the curve to the left.

What do you think happens when we subtract a number from the argument (such as occurs in your problem)?

anonymous · Answer

Shifts the curve to the right?

whpalmer4 · Answer

That's correct!

whpalmer4 · Answer

So, if our original function is $$y=f(x) = x^2$$ and we change it to $$y_{new} = f(x-3) = (x-3)^2$$the new graph looks like?

anonymous · Answer

Idk how to draw it lol

whpalmer4 · Answer

Well, describe what happens to it...

whpalmer4 · Answer

if you don't know what $y = x^2$ looks like, make a little table as I did:

x     y=x^2
-4    16
-3       9
-2        4
-1       1
0        0
1        1
2        4
3        9
4        16

and plot those points and connect them with a smooth curve.

whpalmer4 · Answer

then apply the change that subtracting 3 from the argument does (shifting it right or left) to figure out which of the answer choices is correct.

Or, make a table that computes (x-3)^2 directly and plot those points:
x    y = (x-3)^2
...
0      9
1      4
etc.

anonymous · Answer

Call me stupid but uhm, I'm confused. So bad. This is TOO advanced for me xD.

whpalmer4 · Answer

no, review what we said and did:

adding a number to the result $(y=f(x)+1)$ shifted the graph up.
subtracting a number from the result $(y = f(x)-1)$  shifted the graph down.

we aren't doing that here, so if any of the answer graphs showed the parabola shifted up or down, they would not be candidates to be a correct answer.  however, it doesn't look like we have any of those, so we haven't accomplished any narrowing of the choices yet :-)

adding a number to the argument of the function $(y=f(x+1))$ shifted the graph to the left.  You can think of it this way: if there's some spot on the original graph that happens at $x = 0$, the same spot has to be at $x+1=0$ or $x = -1$ on the shifted graph.

subtracting a number from the argument of the function $(y = f(x-1))$ shifts the graph to the right.  Again, something that happens at $x = 0$ on the original now has to be at $x-1 = 0$ or $x = 1$.

whpalmer4 · Answer

So, our function is $y = f(x) = x^2$.  We change it to $y = f(x-3) = (x-3)^2$.  That shifts the graph 3 units to the right.  All you have to do is figure out which of those graphs represents the graph of $y = x^2$, shifted 3 units to the right.

whpalmer4 · Answer

You can do it!  Unfortunately, I have to leave now to take my cat to the vet :-(