Graph each function over a one-period interval.
y = -4 s...

Question

Graph each function over a one-period interval.

y = -4 sin(2x - π)

Accepted Answer

Hello. Today we are going to be graphing one period of the given trigonometric function. We are asked to graph one period of Y is equal to negative two multiplied by sine of four X minus two pi. In order to graph this function, there are four pieces of information. We need, we need the graph's amplitude, the period of the function, the phase shift and the vertical shift. In order to obtain these pieces of information, we can compare the giving equation to the general equation. Y is equal to a multiplied by sine of B multiplied by X minus C plus D. Using this general function. Let's go ahead and start by finding the graphs amplitude. Now the amplitude is defined as the absolute value of A. This is where A is the coefficient in front of the sine function. If we take a look at our given trigonometric function, the coefficient in front of the sine function is negative too. What this means is that the amplitude is defined as the absolute value of negative two which will simplify to give us two. What this means is that the maximum and minimum values of our function are going to be from positive two to negative two. Or in other words, the range of the function is going to be from negative two to positive two. Now that we have the amplitude, let's go ahead and find the period of the given function. Now the period for a sine function is defined as two pi divided by the absolute value of B. This is where B is the coefficient in front of the X minus C term. Now, in our given trigonometric function, we don't have this B coefficient. However, in order to obtain the B coefficient, we can factor out a common factor from four X minus two pi. Now we want a coefficient of one in front of the X term. In order to obtain this, we want to factor out of four from four X minus two. Pi factoring out this four will allow us to rewrite the function as Y is equal to negative two, multiplied by sine of four, multiplied by X minus pi divided by two. So the coefficient that we factored out is four, meaning that B will be defined as four for a given function. That means the period will equal to two pi divided by the absolute value of four, which will simplify to give us pi divided by two. So since the period is pi divided by two, that tells us that the domain for a given sine function is going to start at zero and end at pi divided by two. Now that we have the period, let's go ahead and find the face shift. Now, the phase shift can also be thought of as a horizontal shift of the graph. The phase shift is defined by the C variable in the general function. If we take a look at our given function, the C variable is equal to pi divided by two. What this means is that we have a phase shift of pi divided by two. That means that every point on the given graph is going to move to the right pi divided by two units. Finally, we need to find a vertical shift. Now, the vertical shift is how many units up or down that we are going to move the graph. And the vertical shift is defined by the D term of the general function. Now the D term is any additional values that we are adding or subtracting to the given function, we are not adding or subtracting any additional values to this function. So that means that the D term will equal to zero. That means we have no vertical shift of the graph. So now that we have the four pieces of information that we need. Let's go ahead and start graphing this function. So right now we know that the period of the function is pi divided by two. And we know that the amplitude is two for a sign function. A sine function always starts at the origin has a maximum value at one has a zero value halfway through the graph, a minimum value and negative one. And, and add to pi traditionally, in this case here, since the period is pi divided by two, we will be condensing one rotation of the sine function to just pi divided by two. Now, since the amplitude is two, our maximum and minimum values are going to reach to positive and negative two. Furthermore, we have a phase shift of pi divided by two. That means that this entire function is going to shift pi divided by two units to the right. However, our period is also pi divided by two. That means the phase shift will not affect the shape of our function. Now, finally, there is one more transformation we need to keep in mind our function has a negative coefficient. This means that we'll be rotating every point of the function with respect to the X axis doing so will allow us to change the shape to be a negative sine function. And this is going to be the rough shape of the given trigonometric function with respect to one period. So I hope this video helps you in understanding how to graph this type of function. And I'll go ahead and see you all in the next video.

Graph each function over a one-period interval.
y = -4 sin(2x - π)

Key Concepts

Sine Function

Amplitude and Vertical Shift

Phase Shift and Frequency

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