Find the period and sketch the graph of the equation. Show the asymptotes. y=-3 tan((1)/(3) x-(π

Find the period and sketch the graph of the equation. Show...

Key Concepts

Tangent Function Period

The period of the tangent function is the length of one complete cycle and is given by the formula ?/|b| for a function of the form tan(bx). When the tangent function is horizontally scaled, this scaling directly affects the period of the function. Understanding the period is crucial for accurately plotting the repeating nature of the tangent curve.

Horizontal Transformation (Phase Shift and Scaling)

Horizontal transformations involve both a scaling (compression or stretching) and a translation (phase shift) of the function. In a tangent function written in the form tan(bx - c), the value b affects the period (as described previously) while the constant c results in a phase shift, moving the graph left or right along the x-axis. These adjustments are key to locating important features such as where the function approaches its asymptotes.

Vertical Transformation (Stretch/Reflection)

Vertical transformations include stretching, compressing, and reflecting the basic tangent function. Multiplying by a constant A scales the output values of the function vertically. When A is negative, it reflects the graph over the x-axis. Although the tangent function doesn’t have an amplitude in the traditional sense, the vertical stretch factor determines the steepness of the curve and influences how quickly the function rises or falls between its asymptotes.

Asymptotes of the Tangent Function

Tangent functions have vertical asymptotes, which are the lines x = (?/2 + k?)/b for integer k, where the function is undefined and heads towards positive or negative infinity. These asymptotes are significant because they mark the boundaries of each period and help in accurately sketching the function’s behavior, showing where the graph will have discontinuities.

Transcript

00:01 We're asked to plot, find the asymptotes and the period of this function here, minus 3 tangent of x over 3 minus pi over 3.

00:13 So to find the period, we need to set, we're scaling the x, x, x is by 1 3.

00:20 So the period of tangent is pi, is pi.

00:23 So the period of this function will be 3 pi because we have x over 3 here.

00:34 So we can see here that i've actually plotted this from minus pi over 2 to positive 3 pi over 2.

00:45 So that's one period.

00:47 Here's one period.

00:49 Well actually i plotted it actually over 2 periods.

00:51 Sorry i plotted it from minus 3 pi to 3 pi because this is two periods of the tangent function now now we have then to find the ascentotes remember that the tangent function tangent equals sine over cosine so the asymptotes are when cosine is zero so that's when x over 3 minus pi over 3 equals pi well let's see here let's do it this way we can say plus or minus plus or minus pi over 2 plus 2 and pi so let's see here we can have and we can rewrite this as let's see this is multiplied through by 3 you get x equals pi plus or minus pi over 2 plus 6 n pi so what does that tell us well if we say n equals 0 we know we have one one one uh one acinote and let's see here when um we know we have one um one one one uh one acetote and let's see here when um we is this right? oh no, i forgot to multiply that by three.

02:37 There, we need a three in there.

02:39 Okay, so let's see.

02:41 When nx0, we have an acentote at minus pi over two...

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