Professor Dave again, let’s transform some functions.
We just learned how to graph some basic functions, like F of X equals X squared.
This is a parabola that opens in the positive direction, with its vertex at the origin.
But functions are rarely this simple.
What if we want to graph X squared plus two?
Or X plus two quantity squared?
Or two X squared?
In each case, we could do one of two things.
We could make a table and find some ordered pairs to plot, or we could recognize how the
number two is transforming the graph of X squared in a different way for each case.
Let’s make some tables and I’ll show you what I mean.
Take the first example, X squared plus two.
Let’s plug in the values we would typically use when evaluating X squared, and we can
see that in each case, we simply add two to what we would have gotten normally.
If the function is two greater, then it sits two units higher on the graph.
So when we add or subtract a number in this way, we are producing a vertical shift, which
can be either positive, shifting up, or negative, and shifting down.
Now that we know how this works, we don’t have to make tables for this kind of transformation
anymore.
If we see X squared minus three, we know what to do.
We just take the X squared graph, and we simply shift it downwards by three, putting the vertex
at zero, negative three.
Now let’s look at the second case, X plus two quantity squared.
Again, let’s make a table so we can see what’s going on.
As we can see, when we plug in negative two, we have to first add two, so it’s like we
are plugging in zero.
Going up from there, we get some more values.
Let’s do a few more to the left as well, and this is what we get.
We can see that it’s the same as X squared, but it has undergone a horizontal shift of
two to the left, with the vertex now at negative two, zero, rather than the origin.
This one is a little trickier to remember than the vertical shift, because a positive
number here produces a shift in the negative horizontal direction.
Likewise, a negative number here, like with X minus two quantity squared, will produce
a shift in the positive horizontal direction.
But if you can memorize this little difference, we have mastered both vertical and horizontal shifts.
Now let’s look at two X squared.
Again, we make a table, and we see that where the function used to equal one, it equals two.
Where it used to equal four, it equals eight.
Graphing these points, we see that it looks like X squared, but it has been stretched
out, so that it is twice as narrow.
This is called a vertical stretch.
Whatever this coefficient is, we stretch by that factor.
If this coefficient is less than one, it will shrink in the opposite direction.
Let’s say this term is one half.
Then the parabola becomes twice as wide as X squared, because it grows at half the rate.
If instead, there is a coefficient inside this term, like two X quantity squared, we
will get a horizontal stretch.
This will be different from the vertical stretch in that if the coefficient is greater than
one, it will shrink horizontally, and if it is less than one, the function will stretch.
Next let’s see what happens if this coefficient is simply negative one, giving us negative
X squared.
Again, making the table, we can see that because we square first and then flip the sign, we
get the same values but they are all negative.
We can plot these, and we can see that we get the graph of X squared, but it is as though
the function has been flipped upside down, or reflected across the X axis.
If instead it was negative X quantity squared, this would produce a reflection across the
Y axis, which in this case will make no difference, but in other instances, like with X cubed,
it will change the graph.
Now let’s look at an example that combines multiple transformations.
What if we have the function negative two times X minus three quantity squared plus four.
We have several transformations to apply here, but we can just do them one at a time, starting
with X squared, and everything will be fine.
First, there is a vertical shift of four, so let’s bump this vertex up four.
Then, there is a horizontal shift of three to the right.
Remember, for the horizontal shift, a negative number here produces a shift in the positive
direction, so we bump the vertex three to the right.
Next, we have a two here.
That means we can stretch this out by a factor of two.
And lastly, there is a negative sign here, so we can reflect this so that it points downwards
instead of upwards.
And that’s all there is to it.
The beauty of interpreting graphs in terms of these types of shifts, is that they apply
to graphs of any type.
Just the way X squared minus two is X squared shifted two units down, we can say the same
thing about absolute value of X, minus two, or X cubed minus two, or the square root of
X, minus two.
Once we know the way that the graphs of common functions appear, it becomes very easy to
graph variations of these curves by applying the transformations we have just learned.
Here is a table summarizing the types of transformations we have just covered, which you can refer
to when graphing.
Of course, sometimes a function will be in a form that can’t be reduced to a series
of transformations, and in this case we need another strategy, so we will go over these
situations later in the series.
For now, let’s check comprehension.