Graphing Tangent Curves

Graphing Tangent Curves:

The curve for y = tan x is very different from the cosine and sine curves.
Since tan x equals the ratio of opposite over adjacent, we find values for tan x
from the unit circle since the y coordinate is sin x and the x coordinate is cos x.
Notice that when cos x = 0, at and at the tangent value is undefined since
we're dividing by zero. Division by zero, though undefined in our number system,
is given the value when the numerator is positive, and when the numerator is negative.
So, whereas .
The tangent curve has asymptotes at those two values. y = tan x is a cyclic curve like those of sin x and cos x,
however, as you can see there is no amplitude.

The period is and the range is R.
The only parts that move are the point of inflection at (0, 0) and the asymptotes
(A point of inflection is one where the curve changes its shape.)

Properties of Tangent Graphs

The transformed tangent rule is f (x) = a tan b (x – h) + k

 period: range: R asymptotes: pt. of infl: (h, k) a > 0, increasing a < 0, decreasing f (x) < k on half the period f (x) > k on half the period

the asymptotes are at the ends of the period, (h, k) is the midpoint of the interval.

Let's graph ,

period = , the point of inflection will be at there will be an interval of on either side of the point of inflection. are the asymptotes.
The curve will rise more steeply than the basic curve since a = 2.  Practice

1) For each of these tangent functions list :
 i) the period ii) the equations of the asymptotes iii) the point (h, k) a) b) c) .

2) Draw the graph of 1c) above. Solutions

1)

 a) i) period = ii) iii) b) i)period = ii) iii) c) i)period = ii) iii) 2)  .

.

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