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, then 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.

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0. Fundamental Concepts of Algebra3h 32m
1. Equations and Inequalities3h 27m
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3. Functions & Graphs2h 17m
4. Polynomial Functions1h 54m
5. Rational Functions1h 23m
6. Exponential and Logarithmic Functions2h 28m
7. Measuring Angles40m
8. Trigonometric Functions on Right Triangles2h 5m
9. Unit Circle1h 19m
10. Graphing Trigonometric Functions1h 19m
11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
12. Trigonometric Identities 2h 34m
13. Non-Right Triangles1h 38m
14. Vectors2h 25m
15. Polar Equations2h 5m
16. Parametric Equations1h 6m
17. Graphing Complex Numbers1h 7m
18. Systems of Equations and Matrices3h 6m
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20. Sequences, Series & Induction1h 15m
21. Combinatorics and Probability1h 45m
22. Limits & Continuity1h 49m
23. Intro to Derivatives & Area Under the Curve2h 9m

10. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

Precalculus

10. 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.

option a

option b

option c

option d

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Verified step by step guidance

Start by understanding the function y = cos(x) - 1. This function 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) is moved down by 1 unit.

Identify key points of the cosine function: at x = 0, cos(x) = 1, so y = cos(x) - 1 = 0; at x = π/2, cos(x) = 0, so y = cos(x) - 1 = -1; at x = π, cos(x) = -1, so y = cos(x) - 1 = -2.

Plot these key points on the graph: (0, 0), (π/2, -1), (π, -2), and continue this pattern for other key points such as (3π/2, -1) and (2π, 0).

Draw the curve through these points, ensuring it follows the typical wave pattern of the cosine function, but shifted down by 1 unit.