Given the following function,
y=tan(x+π/4)
1) determine the interval for the principal cycle.
*(Direction - type your answer in interval notation. Use pi as needed. Use integers or fraction for any numbers in the expression)
2) Then for the principal cycle, determine the equations of the vertical asymptotes
3) the coordinates of the center point
4) the coordinates of the halfway points
(Direction - Type ordered pairs for answers than need them)
THANK YOU.
I assume you can answer the questions for
y = tan(x)
now just shift the x-values left by π/4.
To answer these questions, first, let's understand some properties of the tangent function.
1) The principal cycle of the tangent function occurs in the interval (-π/2, π/2). This means that the graph of y = tan(x) repeats itself in this interval. However, in the given function y = tan(x + π/4), there is a horizontal shift of π/4 units to the left. So, to determine the interval for the principal cycle of the given function, we need to shift the interval (-π/2, π/2) by π/4 units to the left.
Shifting the interval by π/4 to the left, we get the interval: (-π/2 - π/4, π/2 - π/4)
Simplifying, this becomes: (-3π/4, -π/4)
Therefore, the interval for the principal cycle of the given function, y = tan(x + π/4), is (-3π/4, -π/4) in interval notation.
2) Vertical asymptotes occur where the function becomes undefined. For the tangent function, it becomes undefined at odd multiples of π/2.
In the given function y = tan(x + π/4), if x + π/4 is an odd multiple of π/2, the function will become undefined. So, we need to solve the equation:
x + π/4 = (2n + 1)π/2
where n is an integer.
Simplifying the equation, we get:
x = (2n + 1)π/2 - π/4
Therefore, the equations of the vertical asymptotes for the principal cycle of the given function are:
x = (2n + 1)π/2 - π/4, where n is an integer.
3) The center point of the principal cycle is the midpoint between two consecutive vertical asymptotes. In the given function, the midpoint between two consecutive vertical asymptotes occurs when:
(x + π/4) = [nπ/2 + (n+1)π/2]/2
Simplifying the equation, we get:
x + π/4 = (2n + 1)π/4
x = (2n + 1)π/4 - π/4
Therefore, the coordinates of the center point for the principal cycle of the given function are:
(x, y) = ((2n + 1)π/4 - π/4, 0)
4) Halfway points are the points on the graph where the function attains half of its maximum or minimum values, i.e., half of the amplitude. In the case of the tangent function, the amplitude is 1.
The halfway points occur when:
y = 1/2 or y = -1/2
For the given function y = tan(x + π/4), we need to solve the equations:
tan(x + π/4) = 1/2 and tan(x + π/4) = -1/2
We can use the inverse tangent function to solve these equations.
Let's solve the first equation:
x + π/4 = arctan(1/2)
x = arctan(1/2) - π/4
Similarly, solving the second equation:
x + π/4 = arctan(-1/2)
x = arctan(-1/2) - π/4
Therefore, the coordinates of the halfway points for the principal cycle of the given function are:
(x, y) = (arctan(1/2) - π/4, 1/2) and (arctan(-1/2) - π/4, -1/2)
That concludes the answers to all the questions!