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 notice 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 three, 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 three 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 one, then it dips through pi over two, 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 one, I'm actually going to start at a value of negative one. Now this graph is going to go up through pi over two, 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 three pi over two, 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 two, we're going to reach a peak at negative pi, and then we're going to dip down through three pi over two, 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 three 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 three. So if our amplitude is three, that means that rather than our peaks and valleys being at one and negative one, our peaks and valleys are going to be at three and negative three. So we're going to have a peak which is up here at positive three. Those are where the peaks are going to be, and we're going to have valleys which are down here at negative three. So what I can do is I can 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 three, and then we're going to go up through pi over two, and then we're going to reach our peak right about there at pi, and then we're going to go back down through three pi over two, 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 two, we can reach a peak right about there, go down through negative three pi over two, and then we can keep waving as we go to the left. So So this right here is what the graph is going to look like for y is equal to negative three times the cosine of x, and that's the solution to the 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

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Multiple Choice

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

Graphing coordinate

Graph of y = -3 cos(x) showing the cosine function with amplitude -3.

Graphing coordinate

Verified step by step guidance

Step 1: Understand the function y = -3 * cos(x). This is a cosine function with an amplitude of 3, a reflection across the x-axis (due to the negative sign), and a period of 2π (since the coefficient of x inside the cosine is 1).

Step 2: Identify the amplitude. The amplitude is the absolute value of the coefficient of the cosine function, which is |−3| = 3. This means the graph will oscillate between -3 and 3.

Step 3: Determine the period. The period of a cosine function is calculated as 2π divided by the coefficient of x inside the cosine. Here, the coefficient is 1, so the period is 2π.

Step 4: Reflect the graph. Since the coefficient of the cosine function is negative, the graph is flipped vertically. This means the peaks of the cosine wave will be at -3 instead of 3, and the troughs will be at 3 instead of -3.

Step 5: Plot key points. Start at (0, -3), then move to (π/2, 0), (π, 3), (3π/2, 0), and (2π, -3). Repeat this pattern for additional cycles, ensuring the graph reflects the amplitude, period, and reflection.