Graph each function over a one-period interval.
y = 2 + ...

Question

Graph each function over a one-period interval.

y = 2 + 3 sec (2x - π)

Accepted Answer

Hello, everyone. We are asked to graph the following function considering only one period, we are given Y equals five plus four C can of in parentheses, two X minus pi divided by two. We're provided a coordinate plane that shows us the positive side of the X axis and a Y axis that ranges from negative 15 to positive 15. The Y axis is measured in single units but marked every five. The X axis is measured in pi divided by four units. First recall that C can is the reciprocal of cosign. So we're gonna start by rewriting this equation in terms of cosine. So we will have Y equals five plus four cosine of two X minus pi divided by two in parentheses. We use this to find our asthma totes for our C can't function. Remembering that our general format for cosine is that Y equals a cosine of in parentheses. BX minus C closed parentheses plus D. We can start identifying what parts we need to graph. So I'm gonna start by trying to find the period of this graph and recalling that period for cosine is two pi and that we can find our period by doing two pi divided by B B in this case is two. So two pi divided by two means we have a period of pi we need to split our period into four intervals. So I'm gonna start by doing pi divided by four. So making a table my X values cosine starts at the 0.00. So my first X value will be zero. My next will be zero plus pi divided by four, which is pi divided by four from there. I need to do pi divided by four plus pi divided by four could be two pie divided by four which reduces the pi divided by two. Next one. I will take pi divided by two and add pi divided by four considering common denominators. This would get us three pi divided by four. And for my last marked interval, three pi divided by four plus pi divided by four leaves us with pi next to find the Y values to go with that. I recall that cosine starts at 00 and increases to positive one when it is not stretched. So here cosine would still begin at 00. The four in front of the cosine tells us how much we are stretching our graph. So typically why here would be one. But since we have a 41 multiplied by four is four, then I'll decrease back down to zero. Then the next interval typically would be negative one. But again, we're stretching at four. So this will be negative four and then it returns to zero. So we give the points 00, which I'm going to plot as I read pi divided by 44 pi divided by 20, three, pi divided by four, negative four and pi and zero. So this would be one period of our cosine graph. So for our C can graph, we draw asthma totes in where the cosine graph crosses the X axis. So at zero pi divided by two and pi so I put those in as dashed lines. Next, when I draw the C can I make a U shape that shares a point with the cosine function but fits between the asthma toes. So I will have a U shape in this first section between zero and pi divided by two that shares the point pi divided by 44 and opens upward. And then in the interval, pi divided by two to pi, it will have a U shape that opens downward. It'll share the 0.3 pi divided by four, negative four. So we have an upside down new shape. Now there's one more piece we need to take into consideration before we have our final graph. And that's that plus D D here was five and D is our vertical shift. So this means our graph is gonna shift up five places. So in the first interval where we started the point pi divided by 44, we're shift that up five places and we'll end up at the point pi divided by 49. And we'll make our U shape there. And then for our upside down U shape in the interval, pi divided by two pi we are starting at three pi divided by four and negative four. So if we shift that up five places, we'll be working with three pi divided by 41. So I'm redrawing my upside down U shape there. So the black graph on my coordinate plane should be our solution. We have an interval with asthma todes from zero to pi divided by two. On the X, we have a U shape there that has a point at pi divided by 49. And then in the interval, pi divided by two pi we have an upside down U shape with 8.3 pi divided by 41, have a nice day.

Graph each function over a one-period interval.
y = 2 + 3 sec (2x - π)

Key Concepts

Secant Function

Transformations of Functions

Period of Trigonometric Functions

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