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How to Graph 1-cos(x)

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  • 0:00 Steps to Solve 1-cos(x)
  • 2:33 Other Function Transformations
  • 5:23 Lesson Summary
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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.

Steps to Solve 1-cos(x)

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.


Graph of cos(x)
transcos1


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.


Reflect over y-axis
transcos2


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.


Shift up 1 unit
transcos3


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.

Solution

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


transcos4


Other Function Transformations

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).


Shift 5 units left
transcos5


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