Graph each function over a two-period interval.
y = -1 +...

Question

Graph each function over a two-period interval.

y = -1 + 2 tan x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to graph the following function considered two periods. Our function is Y equals negative three plus four tangent of X. Then we have a graph, we have a vertical Y axis, a horizontal X axis. They come together the origin in the middle. The range for what's drawn for our Y axis is from negative 10 to positive 10. And the domain for what's shown for our X axis is from negative three pi divided by four to positive seven pi divided by four. All right. Step one, when we're going to graph a tangent function is we want to identify the period. Our standard tangent function is given to us with the equation Y equals C plus A multiplied by the tangent of B multiplied by the quantity of X minus D. Our period is found by looking at this B value, the period of a tangent function, it is going to be pi divided by B and B is what's being multiplied directly to the X inside of the function here. Nothing is so that B value is one if we have pi divided by A B value of one that's a period of pi. The next thing we want to identify when graphing a tangent function is our vertical asymptotes. The vertical asymptotes are related to a phase shift. And we look at the phase shift by identifying that D value what's being added or subtracted directly to the X inside of the function. Here, nothing is our D value is equal to zero. So that means there is no phase shift. So a standard tangent function as vertical asymptotes at X equals negative pi divided by two and then one period later. So add pi to that at pi divided by two. But since we're considering two periods, we'll add another pi to that pi divided by two and have another vertical Asymptote at three pi divided by two. Now we can draw those. So X equals negative pi divided by two. So our first vertical Asymptote X equals positive pi divided by two. So our second vertical Asymptote and X equals three pi divided by two, our third vertical Asymptote. OK. The next thing that we do when graphing a tangent function is we're going to create an XY table to discern our X values. We want to divide each period into four equal parts. We don't need to figure out the Y values for our vertical asymptotes because those are undefined. But we want to figure out three points in between each vertical Asymptote for a period of pie dividing that into four equal parts, we're going to go in increments of pi divided by four. So from our first vertical Asymptote it negative pi divided by two, we're going to add pi divided by four that gives us an X value of negative pi divided by four. Add a pi divided by four to that. We get an X value of zero, add a pi divided by four to that and we get a positive pi divided by four. And we can see that the graph that we were given already has it in increments of pi divided by four, which is fantastic. But since we are drawing two periods worth of information, we can choose three more X values to plot. So to the right of our vertical Asymptote X equals pi divided by two, we add pi divided by four to that. We get three pi divided by four, add pie divided by four. To that, we get pie and add pie divided by four to that, we get five pie divided by four. OK? Now we want to plug in those X values into our function which is Y equals negative three plus four tangent of X. So we put in the X value negative pi divided by four into that equation. And you can use your calculator, make sure that you're in radiance mode and you'll get a Y value of negative seven, plug in zero for the X value for our negative three plus four tangent of X and you'll get a Y value of negative three plug in pi divided by four for the X value and you'll get a Y value of positive one. Now you can continue to plug in X values to solve for the second period, but the Y values are actually just going to repeat. So for three pi divided by four, the Y value will be negative seven for an X value of pi, the Y value will be negative three. And for an X value of five pi divided by four, the Y value will be one. So now let's graph those negative pi divided by negative pi divided by four, negative seven is going to be here zero, negative three there and pi divided by 41. It's here three pi divided by four negative seven. Here pi divided by or excuse me, pi negative three is right here and Phi Pi divided by 41 is here. The last thing we need to do is connect these as smoothly as possible. We can see that they are going up so immediately to the right of a vertical Asymptote X equals negative pi divided by two. Our function line is heading down toward negative infinity for the Y values along that Asymptote and it curves and it comes up, it's going to go through our point negative pi divided by four, negative seven. It's going to continue to curve and come through our point zero. Negative three, come through a point pi divided by 41 and then it's going to curve and head up toward positive infinity along the vertical Asymptote X equals pi divided by two. None. That same shape is going to repeat itself for the second period to the right of our vertical Asymptote X equals pi divided by two. The function line is heading down toward negative infinity for the Y values. Then it curves and it comes up passes through the 0.3 pi divided by four, negative seven passes through the point pi negative three passes through the 0.5 pi divided by 41. Then curves and comes up heading toward positive infinity for the Y values along the vertical Asymptote X equals three pi divided by two. And that's it. There's the graph of our function well done. We'll catch you on the next one.

Graph each function over a two-period interval.
y = -1 + 2 tan x

Key Concepts

Period of the Tangent Function

Vertical Asymptotes of Tangent

Transformations of Trigonometric Functions

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