06Gtangraph

= Tan Graph =

Recall, for all trig graphs
 * the ** median line ** is a horizontal line through the centre of the graph
 * the ** amplitude ** is the maximum height above the median line
 * the ** period ** is the distance in the x-direction to complete one cycle

If we graph y = tan(x) between –p /2 and p /2, we get:



Notice that:
 * tan(x) approaches positive infinity as x approaches p /2
 * imagine a right-angled triangle with q approaching p /2
 * we have a vertical asymptote at x = p /2
 * tan(x) approaches negative infinity as x approaches –p /2
 * we have a vertical asymptote at x = –p /2
 * tan(x) passes through the origin : . ( 0, 0 )
 * tan(x) passes through ( p /4, 1 ) and ( – p /4, –1 )
 * the shape has a rotational symmetry (about the origin)


 * the __**median line**__ is at y = 0
 * the __**amplitude**__ has no meaning (amplitude is infiinity)
 * the __**period**__ is p


 * If we repeat this shape multiple times we get the full tan(x) graph:


 * y = tan(x) **


 * tan(x) is a cyclic function that repeats its to infinity
 * (in both positive and negative directions)
 * the x-intercepts are at all the integer multiples of p
 * the vertical asymptotes are at the odd number multiples of p /2
 * tan(x) is defined for all of R excluding the odd number multiples of p /2
 * tan(x) is positive in Q1 and Q3 and negative in Q2 and Q4



For the graph, y = tan(x) math \\.\\ .\qquad \text{x-intercepts : } \; x = k\pi \; k \in Z \quad \big\{ \text{Integers} \big\} \\.\\ .\qquad \text{Asymptotes : } \; x = \dfrac{ \big(2k+1\big) \pi}{2}, \; k \in Z \\.\\ math math . \qquad \text{Domain : } \; x \in R \backslash \left\{ \dfrac{ \big(2k+1\big) \pi}{2} \; k \in Z \right\} \\.\\ . \qquad \text{Range : } \; y \in R \\.\\ . \qquad \text{passes through the point } \; \left( \dfrac{\pi}{4}, \; 1 \right) math
 * median line: y = 0
 * no amplitude
 * period = p


 * Dilations : y = atan(nx) **

y = ** a ** tan(x) causes math . \\ . \qquad \qquad \text{Point } \; \left( \dfrac{\pi}{4}, \; 1 \right) \text{ moves to } \; \left( \dfrac{\pi}{4}, \; a \right) \\. math
 * a dilation by a factor of ** a ** in the y-direction
 * if a > 1, the graph is stretched vertically from the median line
 * if a < 1, the graph is compressed vertically towards the median line

y = tan( ** n ** x) causes math . \\ . \qquad \qquad \text{Point } \; \left( \dfrac{\pi}{4}, \; 1 \right) \text{ moves to } \; \left( \dfrac{\pi}{4n}, \; 1 \right) \\.\\ . \qquad \qquad \text{Asympotes move to : } \; x = \dfrac{ \big(2k+1\big) \pi}{2n}, \; k \in Z \\.\\ . \qquad \qquad \text{Period changes to : } \; \dfrac{\pi}{n} \\.\\ . \qquad \qquad \text{x-intercepts change to :} \; x = \dfrac{k\pi}{n} math
 * a dilation by a factor of ** 1/n ** in the x-direction
 * if n > 1, the graph is compressed horizontally towards the y axis
 * if n < 1, the graph is stretched horizontally from the y-axis[[image:06Gperiod.gif align="right"]]


 * Example 1 **

... ... Sketch y = 2tan(0.5x)


 * Solution:**

math . \qquad \text{Median line } y = 0\\.\\ . \qquad \text{X-intercepts } x = \dfrac{k\pi}{0.5} = 2k\pi, \quad k \in Z\\.\\ . \qquad \text{Period } = \dfrac{\pi}{0.5} = 2\pi \\.\\ . \qquad \text{Asymptotes : } x = \dfrac{ \big(2k+1\big) \pi}{2(0.5)} = \big(2k+1\big)\pi, \quad k \in z \\.\\ . \qquad \text{Point : } \; \left( \dfrac{\pi}{4(0.5)}, \; 2 \right) = \left( \dfrac{\pi}{2}, \; 2 \right) math




 * Vertical Translations: y = tan(x) + c **

y = tan(x) + ** c ** causes
 * a translation (shift) up (or down) by a distance of ** c **


 * the median line moves up (or down) to ** y = c **


 * the graph will intercept the median line where tan(x) = 0
 * ie at integer multiples of p


 * the graph will have x-intercepts at the solutions to tan(x) + c = 0

math . \qquad \qquad \text{Point } \; \left( \dfrac{\pi}{4}, \; 1 \right) \text{ moves to } \; \left( \dfrac{\pi}{4}, \; c \right) math


 * Example 2 **

... ... Sketch y = tan(x) + 2 .. for x Î [ – <span style="font-family: Symbol,sans-serif; font-size: 120%;">p, <span style="font-family: Symbol,sans-serif; font-size: 120%;">p ]


 * Solution:**




 * Note: **
 * x-intercepts will be found by solving tan(x) + 2 = 0
 * add integer multiples of <span style="font-family: Symbol,sans-serif; font-size: 120%;">p to the first answer to find all the solutions in the domain

math . \qquad \tan \big( x \big) + 2 = 0 \\.\\ . \qquad \tan \big( x \big) = -2 \\.\\ . \qquad x = \tan^{-1} \big( -2 \big) \\.\\ . \qquad x \approx -1.11^c \quad and \quad x \approx -1.11 + \pi \\.\\ . \qquad x \approx -1.11^c \quad and \quad x \approx 2.03^c math

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