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

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 one 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 two, and then we're going to reach a valley when we get to pi, and then this graph is going to go back up, and it's gonna pass through three pi over two, 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 is going to pass through negative pi over two, and then we're going to reach a valley at negative pi, and this is gonna come back up through negative three pi over two 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 one, 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 one unit, meaning both of these valleys that we see at an output of negative one, these are going to go down here to negative two on the y axis. Likewise, this peak that we had a positive one is going to go down to zero because we shifted down by one unit, and then these points here that we have at a y value of zero, 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 two. And 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 two. 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 one 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, and 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.

12. Trigonometric Functions

Graphs of Sine & Cosine

Business Calculus

12. Trigonometric Functions

Graphs of Sine & Cosine

Struggling with Business Calculus?

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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

Step 1: Understand the function y = cos(x) - 1. This function is derived from the standard cosine function y = cos(x), but it is shifted downward by 1 unit. This means the entire graph of cos(x) is translated vertically.

Step 2: Identify the key points of the cosine function. The cosine function has a period of 2π, with key points at x = 0, π/2, π, 3π/2, and 2π. At these points, the values of cos(x) are 1, 0, -1, 0, and 1 respectively.

Step 3: Apply the vertical shift. Subtract 1 from each of the cosine values at the key points. For example, at x = 0, cos(0) = 1, so y = cos(0) - 1 = 0. Similarly, at x = π, cos(π) = -1, so y = cos(π) - 1 = -2.

Step 4: Plot the adjusted points on the graph. For example, at x = 0, y = 0; at x = π/2, y = -1; at x = π, y = -2; at x = 3π/2, y = -1; and at x = 2π, y = 0. Connect these points smoothly to form the graph.

Step 5: Verify the graph's shape and periodicity. The graph should resemble the cosine function but shifted downward by 1 unit. It should repeat every 2π, maintaining the same amplitude and frequency as the original cosine function.