In this video, we will graph
two absolute value functions,
when the equations are in the form of y or f of x
equals the absolute value of the quantity x minus h plus k,
where the value of h will shift the graph left or right,
and the value of k will shift the graph up or down.
When h is positive or greater than zero,
inside the absolute value we have the quantity x minus h,
so when we have subtraction h is positive,
and the graph is shifted right h units.
And when h is less than zero or negative,
inside the absolute value we would have
x minus negative h, which simplifies to x plus h,
so when we have addition inside the absolute value,
h is negaitve and the graph is shifted left
the absolute value of h units.
And then for k, if k is positive,
the graph is shifted up k units.
Notice how k is outside the absolute value.
And when is negative or less than zero,
the graph is shifted down the absolute value of k units.
So looking at our first equation we have y or g of x
equals the absolute value of the
quantity x plus two minus three.
So because we have this plus two inside the absolute value
and in standard format, has to be subtraction.
We need to think of this as x minus negative two.
Which means h is negative two and therefore the graph
is going to be shifted left the absolute value
of negative two units or left two units.
And then because we have a minus three
outside the absolute value, k is negative three,
which means the graph is going to be shifted down,
the absolute value of negative three units,
or down three units.
And because the vertex is h comma k,
we also know the vertex is the ordered pair,
negative two comma negative three.
Before we graph this by hand, let's verify
this transformation using animation.
Again based upon our equation we now know,
h is negative two and k is negative three.
So notice when h is negative two,
the graph is shifted left two units,
and when k is negative three,
the graph is shifted down three units.
Which means to graph the given function,
we need to take the points on the parent,
or basic absolute value funciton,
and shift them left two, and down three.
So going back to our graph let's start by translating
the vertex of the basic absolute value function.
So we will take this point here and then we will
shift it left two units and down three units.
And therefore the vertex for the given
absolute value function is this point here.
And now let's do that for two more points on the parent
absolute value function, let's use this point here
to the right of the vertex and again we will now shift it
left two and down three, which gives us this point here.
And let's do one more point, let's say this point here
to the left of the vertex, and again left two, down three.
And we now have a point on
the given absolute value function.
Notice how we have enough points to make nice v shape,
using these three points, so the graph of the given
absolute value function is this graph here.
This is the right side and this is the left side.
Let's look at another example, here we have y or g of x
equals the absolute value of
the quantity x minus three plus two.
Because we have minus three inside the absolute value,
we know h is going to be positive, h is positive three.
Which means the shift is going to be right three units.
And because we have a plus two outside the absolute value,
k is positive two and therefore
the shift is also up two units.
And because h is three and k is two,
we know the vertex is three comma two.
And again let's verify this by going back to the animation.
Based upon our equation we know h is three,
notice when h is three the graph
is shifted right three units.
And when k is two the graph is also shifted up two units.
This is a graph of the given absolute value function.
Now we will take points on the basic or parent
absolute value function and shift them
right three and up two units.
So again let's start with the vertex.
And we will shift the vertex right three and up two.
Which gives us the vertex of
the given absolute value function.
Which notice how it does have
the ordered pair three comma two.
And now we will pick two more points,
one on the left and one on the right
of the vertex of the parent function.
Starting with this point here we
shift the point right three and up two,
to find a point on the given absolute value function.
And let's also pick a point to
the left of the vertex on the parent function.
Let's say this point here, right three, up two.
And by using these three points we can make
a nice graph of the given absolute value function.
We know the graph is going to be a v shape
passing through these three points.
Which means to the right of the vertex,
this would be the graph,
and to the left of the vertex, this is the graph.
I hope you found this helpful.