We want to graph the absolute value function, y or g of x
equals negative 1/2 times the absolute value
of the quantity x minus 3 plus 4.
This equation is the standard form
of an absolute value function,
which is y equals a times the absolute value
of the quantity x minus h plus k
where the values of a, h and k transform the graph
of the parent or basic absolute value function, shown here.
Let's begin by determining the values of a, h and k,
and describing the type of transformation.
We'll notice how a is equal to negative 1/2,
so because a is negative,
we have a reflection across the x axis,
and because the absolute value of a is positive 1/2,
we have a vertical compression by a factor of 1/2
which will make the graph wider.
Notice inside the absolute value, the form is x minus h,
and we have x minus three,
and therefore h is positive three.
So because h is positive three,
we have a shift of right three units.
and then finally we notice that k is equal to positive four.
Because k is positive four,
we also have a shift up four units.
So to summarize, a is equal to negative 1/2,
which gives a reflection across the x axis
and a vertical compression by a factor of 1/2,
and the graph opens down.
Again, the graph opens down because of the reflection,
or because a is negative.
Because h is positive three,
we have a shift right three units,
and because k is four, we have a shift up four units.
And the vertex is three comma four.
Let's verify the transformation, using graphing software,
and then we'll come back and graph it by hand.
Let's first change the value of a to negative 1/2,
which will reflect the graph across the x axis,
and also perform a vertical compression by a factor of 1/2.
So, if we take a look at this point
on the parent function, which is two comma two,
if we multiply the y coordinate of two by a negative 1/2,
two times negative 1/2 is negative one,
which gives us a corresponding point after the reflection
and vertical compression, which is two comma negative one.
And notice how this point here is negative two comma two,
again, if we multiply the y coordinate of two
by negative one half, that gives us negative one,
the corresponding point is two comma negative one,
at this point here.
And now let's change h to three, notice when h is three,
the graph is shifted right three units.
And then finally, k is positive four,
which will shift the graph up four units.
Here's the final graph of the given absolute value function.
So going back to our work, let's now graph this by hand.
Let's first perform the reflection across the x axis,
and the vertical compression by a factor of 1/2,
by multiplying the y coordinates of the points on the basic,
or parent absolute value function shown here,
by a negative 1/2.
We'll notice how the vertex is at the origin,
or, zero comma zero.
If we multiply the y coordinate of zero by negative 1/2,
we still have the ordered pair of zero comma zero.
So, this point is the vertex after
the reflection and vertical compression.
And to avoid fractions, let's use points
where the y coordinates are multiples of two,
let's use two comma two, and negative two comma two.
Again, we multiply the y coordinates by negative 1/2,
two times negative 1/2 is negative one,
and therefore the corresponding point
after the reflection and vertical compression
is going to be two comma negative one,
which is this point here.
And then using this point, two times negative 1/2
is also negative one, giving us a corresponding point
of negative two comma negative one,
which is this point here.
So, this would be the graph
after the reflection and vertical compression.
Notice how it only takes the vertex
and one point on the left and right of the vertex
to make a nice graph after the transformation.
And now we'll take these three key points
and shift them right three units, and up four units.
Let's go ahead and start with our vertex,
we will shift the vertex right three, and up four.
This is the vertex of the graph
of the given absolute value function.
And now let's shift this point right three and up four.
So, right three and up four.
And let's also shift this point right three and up four.
And now we have enough points to make a nice graph
of the given absolute value function.
And to avoid any confusion, let's erase the green function.
Our final graph is the blue function, which looks like this.
To verify this graph, we can always make a table of values,
which I've done here on this slide.
You may want to pause the video and verify that these points
are on the graph of our final absolute value function.
I hope you found this helpful.