What are the period and 2 consecutive asymptotes?
1. y= -3tan pi x
2. y= 2 sec 4 x
3.y= csc x/3
4.y = 3cot pi x/3
To find the period and 2 consecutive asymptotes for each equation, we need to analyze the given trigonometric functions.
1. The equation is y = -3 tan (πx).
- Period: The period of the tangent function is π, so the period of this function is also π.
- Asymptotes: The tangent function has asymptotes at odd integer multiples of half the period. In this case, the asymptotes occur at x = (2n + 1)(π/2), where n is an integer. Therefore, the consecutive asymptotes of this function are x = (2n + 1)(π/2), where n is an integer.
2. The equation is y = 2 sec (4x).
- Period: The period of the secant function is 2π divided by the absolute value of the coefficient of x. In this case, the coefficient is 4, so the period is 2π/4, which simplifies to π/2.
- Asymptotes: The secant function has asymptotes at even integer multiples of the period. In this case, the asymptotes occur at x = n(π/2), where n is an integer. Therefore, the consecutive asymptotes of this function are x = n(π/2), where n is an integer.
3. The equation is y = csc (x/3).
- Period: The period of the cosecant function is 2π multiplied by the absolute value of the coefficient of x. In this case, the coefficient is 1/3, so the period is 2π * (1/3), which simplifies to 2π/3.
- Asymptotes: The cosecant function has asymptotes at odd integer multiples of the period. In this case, the asymptotes occur at x = (2n + 1)(2π/3), where n is an integer. Therefore, the consecutive asymptotes of this function are x = (2n + 1)(2π/3), where n is an integer.
4. The equation is y = 3 cot (πx/3).
- Period: The period of the cotangent function is π divided by the absolute value of the coefficient of x. In this case, the coefficient is π/3, so the period is π/(π/3), which simplifies to 3.
- Asymptotes: The cotangent function has asymptotes at odd integer multiples of the period. In this case, the asymptotes occur at x = (2n + 1)(3), where n is an integer. Therefore, the consecutive asymptotes of this function are x = (2n + 1)(3), where n is an integer.
In summary:
1. Period: π
Asymptotes: x = (2n + 1)(π/2)
2. Period: π/2
Asymptotes: x = n(π/2)
3. Period: 2π/3
Asymptotes: x = (2n + 1)(2π/3)
4. Period: 3
Asymptotes: x = (2n + 1)(3)
To find the period and the asymptotes of the given trigonometric functions, let's examine each function individually.
1. y = -3tan(pi x):
To determine the period of the tangent function, we need to find the period of the angle inside the tangent function. In this case, the angle is pi x. The period of the tangent function is equal to π divided by the coefficient of x, which is π.
Therefore, the period of y = -3tan(pi x) is 1.
To find the asymptotes, we consider the values of x where the tangent function is undefined, which are the angles that cause the tangent function to approach infinity. In this case, the tangent function is undefined when pi x is equal to odd multiples of π/2.
So, the asymptotes of y = -3tan(pi x) are x = (2n + 1)π/2, where n is an integer.
2. y = 2sec(4x):
To find the period of the secant function, we calculate the period of the angle inside the secant function. Here, the angle is 4x. The period of the secant function is π divided by the coefficient of x inside the function, which is 4.
Hence, the period of y = 2sec(4x) is π/4.
The asymptotes of the secant function occur when the secant function is undefined, which happens when 4x is equal to odd multiples of π/2.
So, the asymptotes of y = 2sec(4x) are x = (2n + 1)(π/8), where n is an integer.
3. y = csc(x/3):
The sine function has a period of 2π. To find the period of y = csc(x/3), we divide the period of the sine function, 2π, by the coefficient of x inside the sine function, which is 3.
Therefore, the period of y = csc(x/3) is 2π/3.
The asymptotes of the cosecant function occur when the sine function is equal to zero. They can be found by setting x/3 to equal the multiples of π.
So, the asymptotes of y = csc(x/3) are x = 3nπ, where n is an integer.
4. y = 3cot(πx/3):
To find the period of the cotangent function, we divide the period of the tangent function, which is π, by the coefficient of x inside the cotangent function, which is 3.
Thus, the period of y = 3cot(πx/3) is π/3.
The asymptotes of the cotangent function occur when the tangent function is equal to zero. To find these asymptotes, we set the angle, πx/3, equal to odd multiples of π/2.
Therefore, the asymptotes of y = 3cot(πx/3) are x = (2n + 1)(π/2), where n is an integer.
In summary:
1. Period = 1, Asymptotes: x = (2n + 1)π/2
2. Period = π/4, Asymptotes: x = (2n + 1)(π/8)
3. Period = 2π/3, Asymptotes: x = 3nπ
4. Period = π/3, Asymptotes: x = (2n + 1)(π/2)