Graphing Tangent Curves |

**Graphing Tangent Curves:**

The curve for *y* = tan ** x** is very different from the cosine and sine curves.

Since tan

from the unit circle since the

Notice that when cos

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

however, as you can see there is no amplitude.

The **period** is and the range is ** R**.

The only parts that move are the

(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)

.

.

*(all content of the MathRoom Lessons **© Tammy the Tutor; 2004 - ).*