Welcome to a practice SAT Math question.
In this question, we're asked which of the following
would shift f of x right three units?
We want to be able to recognize the
type of translation using function notation.
So for this question,
we'll review some notes on function translations
and then we'll also show
how to use the graphing calculator to solve this problem.
Let's first review our notes.
In these notes, we're assuming c is always positive.
So if we have y equals f of the quantity x plus c,
so the input is x plus c, then f of x will be shifted
left c units.
And this is the opposite direction than we might think,
because we're adding a constant to the input x
and the graph is actually moving left.
And if we have y equals f of the quantity x minus c,
again c is positive,
then the graph would be shifted right.
Again, the opposite direction than we might think
because we're subtracting a constant from the input.
Notice how we if we have plus or minus c
inside the parentheses, or part of the input,
then the graph is shifted left or right.
Remember our question talks about a shift right,
so we're looking for a function in the form
of f of the quantity x minus a constant.
But before we go back to our question,
let's also review vertical shifts.
If we have y equals f of x plus c,
notice how the plus c is not part of the input,
then f of x is shifted up c units.
This seems much more intuitive.
Remember f of x is equal to y,
so we're adding a constant to the function value
or y value, which shifts the graph up.
And then if we have y equals f of x minus c,
then the graph is shifted down.
So going back to our question,
and because f of x is shifted right three units,
we should be able to narrow down our choices to b or c,
and while it may seem f of the quantity x plus c
would shift the graph right,
it's actually f of the quantity x minus three.
So our answer is c, f of the quantity x minus three.
Now let's verify this answer with a graphing calculator.
To do this, we're going to enter a basic function at y one,
and then we'll enter one of our choices into y two.
So let's press y equals -
let's use y equals x squared as our basic function.
To make sure we can tell the functions apart,
I'm going to go to the far left, press enter once,
so that y one is going to be the thick graph,
and now for y two, let's go ahead and check our answer.
We're going to enter f of the quantity x minus three
but on the calculator,
it will be y one of the quantity x minus three.
So we're going to press vars, right arrow,
enter, enter, then in parentheses, x minus three.
So the only difference here is
instead of using f, we're using y one.
And to make sure we have the standard window,
we'll press zoom six.
So there's the basic graph,
and here's a graph of the translated function.
Notice how it has been shifted right three units
verifying our solution.
Let's also check a few of these other ones.
If we wanted to check b,
we could press y equals, and just change this
to y one of x plus three
and press graph, and there's the original,
and notice that now the graph is shifted left three units,
and we're looking for a shift right.
Let's also check d and e.
So we'll press y equals,
go down to y two,
and we're going to close the parenthesis here,
so y one of x is f of x, and then we'll add three.
What we'll notice is d would shift the graph up three units,
which we see here.
Now if we go back and change the plus three to minus three,
the graph will be shifted down three units.
So there's the original,
and there's the shift down three units,
which would be from e.
Let's also check a, even though we should recognize this
as a vertical compression.
So go back to y equals,
go ahead and clear this out.
Now I'm going to type in, in parentheses,
one divided by three for one third,
and then vars, right arrow,
enter, enter, for y one and press graph.
Notice how the graph is going to be wider,
because it's vertically compressed.
So if we're not able to recognize
the function translation or transformation
from the function notation,
we can use the graphing calculator
to determine our answer.
I hope you found this helpful.