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High School Precalculus: Help and Review32 chapters | 297 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Let's see how to graph 1 - cos(x). We'll do this using transformations of the function cos(x). After graphing 1 - cos(x), let's dig a little deeper into transformations of functions, and see how useful they are in graphing seemingly complex functions.

To graph 1 - cos(*x*), we need to be familiar with two things. The first is that we need to know what the graph of *y* = cos(*x*) looks like.

We see that the graph of *y* = cos(*x*) looks like a series of hills and valleys, with maximum values of 1 and minimum values of -1. There are some key points labeled as well.

Okay, that's one thing down. The next thing we need to be familiar with is transformations of functions. In mathematics, **transformations of functions** are algebraic manipulations of a function that correspond to transformations of the graph of the function. Notice that to get from cos(*x*) to 1 - cos(*x*), we multiply the function by a negative to get -cos(*x*), then we add 1 to get 1 - cos(*x*). Each of these manipulations correspond to a transformation of the graph of cos(*x*).

There are four different types of transformations of functions, but we will wait to talk a bit more about this until after we graph 1 - cos(*x*). For now, we are only interested in two of the transformations: reflections and vertical shifts.

A **reflection** is a transformation that reflects a function over the *x* or *y* axis. This type of transformation is represented through multiplication by a negative. When we multiply the whole function by a negative, we reflect the graph of that function over the *x*-axis. When we only multiply the *x* variable by a negative, we reflect the function over the *y*-axis. Recall, that to get from cos(*x*) to 1 - cos(*x*), we said that we first multiply cos(*x*) by a negative. We are multiplying the whole function by a negative, so this corresponds to reflecting the graph of cos(*x*) over the *x*-axis.

We now have the graph of -cos(*x*). The next thing we do to 1 - cos(*x*) is add 1. This corresponds to a vertical shift. A **vertical shift** is a transformation that shifts the graph of a function up or down. Algebraically, if we add *c* to a function, we shift the graph of that function up *c* units. Similarly, if we subtract *c* from a function, we shift the graph of that function down *c* units. Since we are adding 1 to -cos(*x*), we will shift the graph up 1 unit.

Alright! All together, we have that to graph 1 - cos(*x*), we start with the graph of cos(*x*), reflect it over the *y*-axis, and shift that graph up 1 unit.

So now that it's solved, take a look at the graph of 1 - cos(*x*), which is shown on your screen now.

As we just saw, transformations of functions come in extremely handy when trying to graph variations of a well-known function. We just saw reflections and vertical shifts in action. Let's take a look at the two other types of transformations of functions. Those are horizontal shifts and stretching/shrinking.

A **horizontal shift** is a transformation that shifts the graph of a function to the right or left. These types of transformations correspond to adding or subtracting a number to or from *x* in a function. If we add *c* to *x* in a function, then we shift the graph of the function left *c* units, and if we subtract *c* from *x* in a function, then we shift the graph of the function right *c* units. That may sound backwards to you, but that is how horizontal shifts work. Just remember, when dealing with horizontal shifts, think 'opposites' (right = subtraction, left = addition).

That's a lot of words! Putting things into practice is always better, so to illustrate this, consider the function *y* = cos(*x* + 5). Since we are adding 5 to *x* in the function cos(*x*), we shift the graph of cos(*x*) five units to the left to obtain the graph of cos(*x* + 5).

**Stretching** and **shrinking** are transformations that either stretch or shrink (compress) a function. Algebraically, this transformation corresponds to multiplying a function or the *x* variable of a function by a number. If we multiply a whole function by *c* or 1/*c*, we stretch or shrink the function vertically, and if we multiply just the *x*-variable in a function by *c* or 1/*c*, we stretch or shrink the function horizontally.

When it comes to vertical stretching and shrinking, if we multiply by *c*, then we are stretching the function vertically by a factor of *c*. If we multiply by 1/*c*, then we are shrinking the function vertically by a factor of *c*.

On the other hand, when it come to horizontal stretching and shrinking, if we multiply *x* by *c*, then we are shrinking the function horizontally by a factor of *c*. If we multiply *x* by 1/*c*, then we are stretching the function horizontally by a factor of *c*.

Oh boy, that's a lot of words again! Let's look at an example again to better understand this concept. Consider the function 2cos(*x*). Since we are multiplying the whole function of cos(*x*) by 2, we are stretching the function cos(*x*) by a factor of 2.

As we saw with our initial problem of graphing 1 - cos(*x*), we can have more than one transformation happening at a time. Consider the function *y* = (1/3)cos(-*x* - 2) - 4. At first glance, this looks very complicated to graph, but really it's just a matter of taking the graph of *y* = cos(*x*), shifting it 2 units to the right, shrinking vertically by a factor of 3, reflecting over the *y* axis, and shifting down 4 units.

Pretty neat, huh?

Let's review what we've learned. In this lesson, we looked at the function 1- cos(x), which is an example of **transformations of functions**, or algebraic manipulations of a function that correspond to transformations of the graph of the function. We looked at four types of transformations, which included **reflections**, which are transformations that reflect a function over the *x* or *y* axis; **vertical shifts**, which are transformations that shift the graph of a function up or down; **horizontal shifts**, which are transformations that shift the graph of a function to the right or left; and **stretching** and **shrinking**, which are transformations that either stretch out or compress a function.

It is easy to see that transformations of functions are extremely useful when trying to graph complex functions. These transformations can take a seemingly difficult problem and transform it into a problem that is quite easy!

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