Welcome to a video on graphing translations
of the basic absolute value function.
In this video, the equations will be
in standard form with a equal to one.
Standard form of an absolute value function
is y equals, or f of x equals a times the absolute value
of the quantity x minus h plus k.
But again, for this video, a is equal to one,
and therefore the equations will be in the form
of y equals the absolute value
of the quantity x minus h plus k.
Before we discuss how h and k affect the graph,
there are a few things we should notice
about the graph of the basic or parent
absolute value function shown here on the left.
Remember, if we're ever not sure how to make a graph,
we can always make a table of values to graph the function.
Notice how the graph of the absolute value function
is a v shape, in this case it opens up.
The lowest point on the graph,
or the point where the graph changes direction
is this point here called the vertex.
Notice how the vertex is at the origin,
which has an ordered pair, zero comma zero.
Notice when x is greater than zero,
or to the right of the vertex,
the graph is linear with a slope of positive one,
meaning, if we start at any point
on the graph on the right side
and go up one and right one,
we can determine another point on the graph.
And then, when x is less than zero,
or to the left of the vertex, the slope is negative one.
Now, going back to the form of the equation
and y equals the absolute value
of the quantity x minus h plus k,
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,
we would have the absolute value of x minus h,
so when we have subtraction here,
h is positive, and the graph is shifted right h units.
And when h is less than zero, or negative,
we would have the absolute value of the quantity
x minus negative h, which simplifies to x plus h.
So, when we have addition here,
h is negative, and the graph is shifted left
the absolute value of h units.
And now, for k, when k is positive, or greater than zero,
the graph is shifted up k units,
and when k is less than zero, or negative,
the graph is shifted down the absolute value of k units.
And then, finally, the vertex is the ordered pair h comma k.
Let's look at an animation to better understand
how the values of h and k affect the graph
of the basic absolute value function.
Let's first see how h affects the graph of the function.
Notice when h is positive,
the graph is shifted right h units,
and when h is negative, the graph is shifted
left the absolute value of h units.
So if we stop here for a moment, where h is negative six,
let's determine the equation of this graph.
Well, if h is negative six, and we know a is one
and k is zero, this would give us the equation
y equals the absolute value
of the quantity x minus negative six.
Simplifying, we have the absolute value
of the quantity x plus six.
So again, when we have addition inside the absolute value,
the graph is shifted left, in this case six units.
And this should make sense because notice how
when x is negative six, the ordered pair here,
the vertex, is negative six comma zero,
and notice when x is negative six,
we do have a zero inside the absolute value,
the absolute value of zero is zero,
and therefore the output, or y value,
is zero when x is negative six.
And when h is positive, let's say when h is positive four,
we would substitute positive four in for h,
which gives us the equation y equals the absolute value
of the quantity x minus four.
Again, when we have subtraction inside the absolute value,
the graph is shifted right in this case four units.
And now let's see how k affects the graph.
So we'll set h back to zero and now change the value of k.
Notice when k is positive, the graph is shifted up k units,
and when k is negative, the graph is shifted down
the absolute value of k units.
So here, where k is equal to negative four,
because we have addition here,
we normally don't write plus negative four,
we normally just write minus four.
And we know a is one and h is zero,
and therefore the equation of this purple graph
is y equals the absolute value of x and then minus four.
And let's take a look at an equation when k is positive.
Let's change k to a positive two.
Notice how the graph is shifted up two units,
and because k is positive two,
the equation of this purple graph, or this graph here,
is y equals the absolute value of x plus two.
Let's take a look at a few more examples.
Let's say you were asked to graph y equals,
or g of x equals the absolute value of x plus three.
And again, because we have addition here,
h is going to be negative three.
We need to be thinking that we can write x plus three
as x minus negative three to attermine the value of h,
again, because in standard form,
we do have subtraction inside the absolute value.
So, because h is negative three,
the graph is shifted left the absolute value
of negative three or three units.
And so notice how the graph would be this orange graph here,
where we have the parent graph,
or basic absolute value function graphed here in blue.
We shift this left three units to form the graph
of y equals the absolute value of the quantity x plus three.
And then to graph y equals the absolute value
of x minus one, because we have subtraction here,
the graph is shifted right one unit
because h is positive one.
So again, the blue graph is
the basic absolute value function.
We would take this graph and shift it right one unit
to graph the given function.
And now let's look at two more examples
involving the value of k.
If we want to graph y equals the absolute value
of x plus four, k is positive four,
which means we shift the basic absolute
value function up four units.
So, in blue we have the basic absolute value function.
Because k is four, we shift the graph up four units
to graph the given function.
And if we have y equals the absolute value of x minus two,
we shift the graph of the parent function,
or the basic absolute value function,
down two units to graph the given function.
So this is an overview of how h and k affect the graph
of the basic absolute value function.
In the next video, we will graph absolute value functions
that have both the value of h and k in the equation.
I hope you found this helpful.