Given f of x equals x squared,
if g of x is f of x shifted up four units
and left one unit, write the formula for g of x.
So if g of x is a transformation of f of x,
we can write g of x in this form here,
for the values of a, b, c, and d,
transform the graph of f of x.
Now for this example, since we only have a shift up
and left, we only have to determine the values of c and d.
So our function g of x is going to be in the form
of g of x equals
f of the quantity x plus c plus d.
Where the value of c shifts the graph left or right.
If c is positive, the graph is shifted left,
and if c is negative, the graph is shifted right.
And the value of d shifts the graph up or down.
If d is positive, the graph is shifted up.
If d is negative, the graph is shifted down.
To get a better feel for this,
let's look at an animation.
So here's the graph of the basic function,
f of x equals x squared.
Again, our goal is to shift this up four units
and left one unit.
Let's start by changing the value of d
to see how it affects the graph.
Notice if d is negative,
or we have minus a constant
after x squared, the graph is shifted down d units.
So if we want to shift up four units,
we'll have to have plus four after the x squared.
So it would look like this,
but now we need to shift the graph left one unit.
So now we'll change the value of c,
and determine how it affects the graph.;
Notice when c is negative, where we have x minus a constant,
the graph is actually shifted right.
This may be the opposite of what we might think,
so we're going to have to x plus a constant squared,
or in this case, the quantity x plus one squared.
Which we see here.
So again, the parent function, or f of x equals x squared
has now been shifted up four units and left one,
so here's our formula for g of x.
G of x will equal the quantity x plus one squared plus four.
And again, this plus four here on the end
shifted the graph up four units,
and the plus one here
shifted the graph left one unit.
So this is the formula that we're looking for,
but sometimes you may be asked to express g of x
in terms of f of x.
So again, if f of x is equal to x squared,
then g of x will have the quantity x plus one squared,
we'd have to have f of the quantity x plus one,
and then for the plus four, we'd have a plus four
after f of the quantity x plus one.
So here's how we can express g of x in terms of f of x,
if asked.
And to finish, let's summarize these
two types of translations.
To shift a graph left or right,
if c is negative or less than zero,
the graph is actually shifted right c units,
so we'd have f of the quantity x minus c.
And if c is positive, or greater than zero,
the graph is shifted left c units,
so we'd have f of the quantity x plus c.
And if the graph is shifted up or down,
if d is negative or less than zero,
the graph is shifted down d units,
so we'd have f of x minus d.
And if d is positive, the graph is shifted up d units,
so we'd have f of x plus d.
Okay, I hope you found this helpful.