Graph the function y = − 3 ⋅ cos ( x ) y=-3\cdot\cos\left(x\right) y = − 3 ⋅ cos ( x ) .

Hey everyone. So let's try solving this problem. Here we are asked to graph the function y is equal to negative three times the cosine of x, and we need to graph it down here. Now to solve this problem, I noticed that we have a couple of different things going on. I noticed that we have a negative sign, which I know is going to reflect the graph over the x axis. And I also see we have a 3, which is going to change the amplitude. Now the way I'm gonna solve this problem, I'm gonna focus on one of these changes at a time, and then eventually build up to our graph, which is y equals negative 3 cosine x. Now the first thing I'm going to do is kind of ignore this amplitude here, and just graph the function for y is equal to the negative cosine of x. So this incorporates both the negative sine and the cosine term. Now I know that the cosine is a graph that starts here at positive 1, then it dips through pi over 2, and then keeps waving that way. But the thing that I can see is that we have negative cosine of x. So that means the graph is going to be flipped over the x axis. So rather than starting at a value of 1, I'm actually gonna start at a value of negative one. Now this graph is going to go up through pi over 2, and then what we're gonna do is we're going to reach a peak as we get to pi. Then likewise, we're going to continue from this peak. We're gonna go down through 3 pi over 2, and then we're going to reach another valley when we get to the end of this. So this is what our cosine graph is going to look like as as we go to the right. And we can extend this graph to the left as well. So we can go up through negative pi over 2, we're going to reach a peak at negative pi, and then we're going to dip down through 3 pi over 2, and this is going to continue waving as we go to the left. So this is what y equals the negative cosine of x is going to look like. Now from here, what I can do is graph the function y is equal to negative 3 times the cosine of x. And the way that I can do that is by recognizing this is a change in the amplitude, and the amplitude is going to change by 3. So if our amplitude is 3, that means that rather than our peaks and valleys being at 1 and negative one, our peaks and valleys are going to be at 3 and negative 3. So going to have a peak, which is up here at positive 3. Those are where the peaks are going to be, and we're going to have have valleys, which are down here at negative 3. So what I can do is I'm gonna recognize this peak will be up here, this peak will also be up here, and this valley is going to be down there. So redrawing this graph, we're going to start at negative 3, and then we're going to go up through pi over 2, and then we're going to reach our peak right about there at pi, and then we're going to go back down through 3 pi over 2, and then this graph will keep waving as we go to the right. And then likewise, as we go to the left, we can go up here through negative pi over 2, we can reach a peak right about there, go down through negative 3 pi over 2, and then we can keep waving as we go to the left. So this right here is what the graph is going to look like for y is equal to negative 3 times the cosine of x, and that's the solution to the 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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6. Exponential and Logarithmic Functions2h 28m
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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

Graph the function $y=-3\cdot\cos\left(x\right)$ .

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Verified step by step guidance

Identify the function to be graphed: y = -3 * cos(x). This is a cosine function with an amplitude of 3 and a reflection over the x-axis due to the negative sign.

Determine the amplitude of the function. The amplitude is the absolute value of the coefficient of the cosine function, which is 3. This means the graph will oscillate between -3 and 3.

Identify the period of the function. The period of a cosine function is given by 2π divided by the coefficient of x inside the cosine function. Since there is no coefficient other than 1, the period remains 2π.

Consider the phase shift and vertical shift. In this function, there is no horizontal or vertical shift, so the graph starts at the origin.

Plot the graph using the identified characteristics: Start at the maximum point (since it's reflected, it starts at -3), then move to the midline at y=0 at π/2, reach the minimum at y=3 at π, return to the midline at 3π/2, and complete the cycle at 2π back at the maximum point.