Sketch the function y = cos ( x ) − 1 y=\cos\left(x\right)-1 y = cos ( x ) − 1 on the graph below.

Hey, everyone. So let's see how we can solve this problem. Here we're asked to sketch the function y is equal to the cosine of x minus 1 on the graph below. Now to solve this problem, what I'm going to start by doing is sketching a function for y is equal to the cosine of x. I'm ignoring this negative one for now because the cosine of x is a graph that I'm familiar with. Because recall from the previous video that the cosine of x starts at a value of positive one on our output. Now along the x axis, what's going to happen is this graph is going to pass through pi over 2. And then we're going to reach a valley when we get to pi, then this graph is going to go back up, and it's gonna pass through 3pi over 2. And then this graph is going to keep waving as we go to the right. Now we can also extend this graph to the left as well. This going to pass through negative pi over 2, and then we're going to reach a valley at negative pi, and this is gonna come back up through negative 3 pi over 2, and keep waving on the left side as well. So This is what the graph for the cosine of x is going to look like. Now in order to graph the function y is equal to the cosine of x minus 1, we can do this by recognizing this minus one is going to cause a shift. It's gonna cause a vertical shift, and because we have a negative number, the graph is going to shift down. So we can take every point and shift it down by 1 unit, meaning both of these valleys that we see at an output of negative one, these are going to go down here to negative 2 on the y axis. Likewise, this peak that we had a positive one is going to go down to 0, because we shifted down by 1 unit. And then these points here that we have at a y value of 0, these are all going to go down to negative one. So if we sketch our new graph, we're going to start at the center of our graph right about there, and then going to the right, this graph is going to dip here, and it's going to reach negative one as an output when we get to pi over 2. And then what we're going to do when we get to pi on the x axis, we're going to reach our valley at an output of negative 2. We're going to cross back up here, and then we're going to touch the x axis again as we move up, because this point has been shifted down 1 unit as well. And likewise, we can extend this to the back of our graph as well. So we're going to cross through here, and then we're going to reach a valley when we get to negative pi, we're going to go back up here and we're going to touch the x axis once again. So this is what the cosine of x minus one graph is going to look like, and that is the solution to our problem. So hope you found this video helpful. Thanks for watching.

4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

Trigonometry

4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

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Multiple Choice

Sketch the function $y=\cos\left(x\right)-1$ on the graph below.

A cosine graph

Verified step by step guidance

Start by understanding the function y = \cos(x) - 1. This is a transformation of the basic cosine function y = \cos(x).

The transformation involves a vertical shift downward by 1 unit. This means that every point on the graph of y = \cos(x) will be moved down by 1 unit.

Recall that the cosine function y = \cos(x) has a maximum value of 1 and a minimum value of -1. With the transformation y = \cos(x) - 1, the maximum value becomes 0 and the minimum value becomes -2.

The period of the cosine function is 2\pi, meaning it repeats every 2\pi units along the x-axis. This property remains unchanged in the transformed function y = \cos(x) - 1.

Sketch the graph by plotting key points: at x = 0, y = \cos(0) - 1 = 0; at x = \pi/2, y = \cos(\pi/2) - 1 = -1; at x = \pi, y = \cos(\pi) - 1 = -2; and continue this pattern to complete the graph over the interval shown.