I need to state the period and 2 consecutive asymptotes on the graph for the following questions.

1: y = -3 tan pi*x
period: pi (?)
asymptotes: ?

2: y = 2 sec 4x
period: ?
asymptotes: ?

3: y = csc (x/3)
period: ?
asymptotes: ?

4: y = 3 cot (pi*x/2)
period: ?
asymptotes: ?

If possible, I'd love some explanation/clarification for the answers as well so I can understand it better... thanks!

1: y = -3 tan pi*x

period: pi (?)
asymptotes: ?
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Well, I will help with the first one. Then you better try to use a similar approach on the rest
y = -3 sin pi x/ cos pi x
this repeats when pi x = pi
or period = 1 pi
It gets undefined when cos pi x = 0
or pi x = pi/2 which is x = 1/2
then pi x = 3 pi/2 which is x = 3/2

Sure! I'd be happy to help you with these questions.

1. For the equation y = -3 tan (πx), the period of the tangent function is π divided by the coefficient of x inside the parentheses. In this case, the coefficient is π, so the period is π/π which simplifies to 1. Therefore, the period of the function is 1.

As for the asymptotes, the tangent function has vertical asymptotes where the angle inside the function approaches odd multiples of π/2. In this case, the angle is πx, so the vertical asymptotes occur when πx approaches odd multiples of π/2. Therefore, the vertical asymptotes are x = -1/2, x = 1/2, x = -3/2, x = 3/2, and so on.

2. For the equation y = 2 sec (4x), the period of the secant function is 2π divided by the coefficient of x inside the parentheses. In this case, the coefficient is 4, so the period is 2π/4 which simplifies to π/2. Therefore, the period of the function is π/2.

The secant function has vertical asymptotes where the angle inside the function approaches odd multiples of π/2. In this case, the angle is 4x, so the vertical asymptotes occur when 4x approaches odd multiples of π/2. Thus, the vertical asymptotes are x = -π/8, x = π/8, x = -3π/8, x = 3π/8, and so on.

3. For the equation y = csc (x/3), the period of the csc function is 2π multiplied by the reciprocal of the coefficient of x inside the parentheses. In this case, the coefficient is 1/3, so the period is 2π/(1/3) which simplifies to 6π. Therefore, the period of the function is 6π.

The cosecant function has vertical asymptotes where the angle inside the function approaches multiples of π. In this case, the angle is x/3, so the vertical asymptotes occur when x/3 approaches multiples of π. Hence, the vertical asymptotes are x = 3π, x = -3π, x = 6π, x = -6π, and so on.

4. For the equation y = 3 cot (πx/2), the period of the cotangent function is π divided by the coefficient of x inside the parentheses. In this case, the coefficient is π/2, so the period is π/(π/2) which simplifies to 2. Therefore, the period of the function is 2.

The cotangent function has vertical asymptotes where the angle inside the function approaches multiples of π. In this case, the angle is πx/2, so the vertical asymptotes occur when πx/2 approaches multiples of π. Therefore, the vertical asymptotes are x = -2, x = 2, x = -4, x = 4, and so on.

I hope this explanation clarifies how to find the period and asymptotes for each of these functions. Let me know if you have any further questions!