Graph each function over a one-period interval.
y = ½ se...

Question

Graph each function over a one-period interval.

y = ½ sec x

Accepted Answer

Hello. For this problem, we are to grasp the following function considering only one period where our function is Y equals 1/4 sin of X. And we're also given a coordinate plane here to graph our equation on. Now, the first thing we should see is that see of X is actually the reciprocal function. We know that seeking of X is equivalent to one divided by cosine X because these are reciprocals of each other. We can make use of the general form of the cosine function. Now, the general cosine form is given by the equation Y equals a cosine BX. In our case, we have C and X which means we have cosine X making our B equals to one. Now, this function tells us our amplitude and our horizontal stretch given by the bee. We can use the value for B to find the period of our function. The equation for our period is given by tie the violent by B. Now we mentioned our B is equals to one. That means our period is two pi divided by one or just two pi when simplified. Now, we can make points from our period to graph on our plot. Let's say we want our first X value since we're not moving to the left or right, our first X value will be zero. Now, a good way to find our points is to take our period and divide it by four. This will give us five equal points to where we can plot our functional with including our X zero equals zero. So we will say our period divided by four is Tupai divided by four which simplifies the pi divided by two. We can now add our period divided by four to each one of our points starting at X zero equals zero. We will say X one is zero plus pi divided by two which is just pi divided by two, X two. We will take our previous value pi divided by two and add another pie divided by two. Thank you. This time X three will be pi plus pi divided by two which gives us three pi divided by two. And finally X four is three pi divided by two plus pi divided by two which gift gives us Tupac. So we now have all of the X values or our points. Now we just need to find our Y values. Now, we did mention before that this is the reciprocal function. So to graph this, we need to make use of our cosine function. We will rewrite our cosine function as Y equals one divided by four cosine X to account for the reciprocal. Now, we just seem to calculate all of our Y values. Let's say our first one is X zero. We have Y equals 1/4 cosine of zero. Now, we know based on our regular circle cosine of zero equals one. So this will be 1/4 multiplied by one, which is just 1/4. You can make a little table here with all of our values, texts and why and we will enter all of our values. Now, we just evaluated that X equals zero gives us a Y of one forts. If we were to do that for every one of our values, we would end up with zero as our next Y value negative 1/4 0 and positive 1/4. Again. Now we have all of our points. We can plot them on our graph. We go up, we will get zero up 1/4 pile of isla by 20. Bye. Negative one fours. Three pi divided by 20, two pie, 1/4. Yeah, we can connect all of these with a smooth curve. Now that just gives us our cosine function. We did mention we wanted our sequent function. Now the second function since it is equivalent to one divided by cosine, we will have asymptotes everywhere where our Y value is equal to zero. I will note those on our graph. We know at pi divided by two and three pie divided by two. Our Y values are zero, which means those are the asymptotes for our seeking graph. No, we just need to draw smooth lines between our asymptotes. We will start at our point. And since our point is at the top of our cosine function here, it's at a peak, we will make a curve that goes upwards like. So that one looks a little strange but we will, we dry it like. So we'll do the same at pi by making a downwards parabola shape and the same at two pi we now have the graph or 1/4 C and X. Ok. I hope to help you solve the problem. Thank you for watching. Goodbye.

Graph each function over a one-period interval.
y = ½ sec x

Key Concepts

Secant Function

Graphing Trigonometric Functions

Amplitude and Vertical Stretch

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