Functions Transformed

Lesson Plan - Get It!

Audio:

A function is essentially a collection of ordered pairs, so it makes sense that applying the transformation to the function would be like applying the transformation to each of the ordered pairs the function defines.

Let's see what that means in more detail!

As you work through this lesson, keep track of your notes with the Function Transformation Notes, found under the Downloadable Resources in the right-hand sidebar.

We started by moving the point (0,0) up by 2. This resulted in an increase in the y-value by 2. In other words, we are taking the y-value and adding 2.

• What if we added 2 to all the points defined by a function, f(x)?

Remember that f(x) itself defines the output, or y-values. Therefore, we can add 2 to all of the outputs or y-values by taking f(x) + 2!

• What do you think would happen if we took f(x) + k, for any positive value of k?
• What about if k is negative?

Let's explore this a bit more with an example.

In the GeoGebra application shown below, Function Transformations: Vertical Shifts, we see the function f(x) = x².

Use the slider to the left to see what happens when you change the value of k in the function g(x) = f(x) + k.

This movement may be fairly intuitive for you -- adding a positive value results in values going up, and adding a negative value results in values going down!

• What happens if we want to shift a function left or right?

In our initial example, we started at the point (0,0) and moved it to the right by 3. This resulted in an increase in the x-value by 3.

You may be thinking this will work out exactly like shifting vertically, and we just need to add 3 to the x-values, or inputs, to shift the function to the right.

Unfortunately for us, horizontal transformations are not quite as intuitive or nice!

• What do you think would happen if we took f(x + 3)?

Let's explore this with an activity.

In the Geogebra application shown below, Function Transformations: Horizontal Shifts, we see the function f(x) = x² again.

Use the slider on the bottom right to see what happens when you change the value of k in the function g(x) = f(x + k).

If this seems backwards, that's ok!

To see a quick explanation for why this is true, check out Explaining Horizontal Shifts of Function from Colette Woodruff Tropp:

To quickly recap the video, we have the graph of the function f(x) = x² and have labeled the point (1,1).

If we shift this function to the left by 1, we will see that the labeled point moves to the point (0,1). This essentially means that we need a smaller input value to get the same output value.

So, if g(x) is the shifted function, we have that g(0) = f(1).

Therefore, g(x) = f(x + 1).

At this point, we have discussed what it means to shift a function up, down, left, and right. These result from adding different values to the output and input of a function.

• What do you think will happen if we multiply the input or output by a number?

Let's explore this with an activity.

In the GeoGebra application shown below, Function Transformations: Vertical Stretches and Compressions, we see the function f(x) = x³ - 2x.

Use the slider to the left to see what happens to the function when we take g(x) = kf(x) for different values of k. In particular, make note of what happens when you set k = 0.5 and k = 2.

It may be a little difficult to see what exactly is happening here, but try to focus on one point on the graph and follow what happens to that point!

Let's look at this further with some pictures.

If we look at the graph of g(x) = 2f(x), it looks like the function has been pulled apart vertically (stretched). Looking at the "bumps" on the graph, we see these have been stretched out:

On the other hand, if we look at the graph of g(x) = 0.5f(x), it looks like the function has been smushed down vertically (compressed). Again, look at the "bumps" on the graph to see this compression:

This last example showed us what happens when we apply vertical stretches and compressions. To do this, we multiply the function by a constant k, which multiplies all of the output values by k.

• What happens if we want to apply stretches and compressions horizontally?

Just like with the shifts left and right, we need to change the value of the inputs, or x-values.

To do this, let's see what happens when we multiply x by k inside the function! In other words, let's explore what happens when we take g(x) = f(kx) for different values of k.

In the GeoGebra application shown below, Function Transformations: Horizontal Stretches and Compressions, we see the function f(x) = x³ - 2x again.

Use the slider on the bottom right to see what happens when you change the value of k. In particular, make note of what happens when k = 0.5 and k = 2.

As we saw before with the horizontal shifts, these horizontal stretches and compressions seem a bit counterintuitive! However, the logic from the previous explanation holds.

We are basically needing different input values to get the same output values as our original function f(x). Whether we need smaller or larger x-values to get the same output determines if the function has been stretched or compressed horizontally.

Make sure you are making note of all of these transformations on your Function Transformation Notes (Downloadable Resources) so you can keep track of this information!

The last transformation we need to discuss is reflecting functions.

At the beginning of the lesson, we explored what happened when we reflected a single point over the x-axis and y-axis. Let's dive in a bit deeper!

Start with the point (1,1) on a coordinate plane as seen below:

If we want to reflect across the x-axis, we can view the x-axis as a mirror. Everything above the x-axis will be mirrored below the x-axis (and vice versa)!

This gives us the point (1,-1):

Note that this reflection made the y-value, or output value, negative! We can represent this using our function notation.

If reflecting across the x-axis makes all of the output values the negative of what they were, then g(x) = -f(x) means we've reflected the entire function, f(x), across the x-axis!

Similarly, we can reflect across the y-axis. This gives the point (-1,1):

This reflection made the x-value, or input value, negative! Therefore, g(x) = f(-x) means we've reflected the entire function f(x), across the y-axis!

Let's see what this looks like for an entire function!

The GeoGebra application shown below, Function Transformations: Reflections, gives the function f(x) = x³ -3x + 1.

On the left-hand side, you can click the circle next to g(x) and h(x) to see what happens if you reflect across the x-axis and y-axis, respectively! Pay attention to each of the "bumps" on the graph to see the reflections!

The function chosen in the example changes each time we reflect it, but this is not true for all functions!

• Can you think of a function that would be the same if you reflect it across the x-axis?
• What about the y-axis?

Hopefully you could think of one that stays the same if you reflect it across the y-axis, but let's explore this together.

Consider the function f(x) = x², graphed below:

• What happens if you reflect this function across the y-axis?

If you said, "it stays the same!", you are absoutely right!

This is a special type of function called an even function. The characteristics of an even function are that they are symmetric across the y-axis, which means that reflecting across the y-axis does not change the function!

This means that f(x) = f(-x)! We can see this for the function f(x) = x².

Another example of an even function is the function y = cos(x):

• What about a function that's the same if you reflect it across the x-axis?

If you try to find an answer to this question, you will run into a problem! This would be the same as asking if there's a function which is symmetric across the x-axis.

However, a graph which is symmetric across the x-axis is not a function!

There is one more special case we have not explored yet! These are special functions which are the same when reflected over both the x-axis and y-axis.

For example, consider the function f(x) = x³, graphed below:

• What happens if you reflect this function across the x-axis and then across the y-axis?

If you said, "it stays the same!", you are absolutely right!

This is a special type of function called an odd function. The characteristics of an odd function are that they are symmetric across the origin, which means that reflecting across the x-axis and y-axis does not change the function!

This means that f(x) = -f(-x)! We can see this for the function f(x) = x³.

Another example of an odd function is the function y = sin(x):

It's important to note that not all functions are classified as even or odd. These are just special types of functions you may encounter!

• Have you been taking notes?

If not, make sure you fill out the Function Transformation Notes (Downloadable Resources) to have a quick reference as you move to the Got It? section!

You can also check your answers with the Function Transformation Notes: ANSWER KEY (Downloadable Resources).