1.HOW do you state the amplitude, period, phase shift and vertical shift for the function y= 2sin[4(angle symbol)-pie/2]-5?
2. Describe the function y= -5sin4 (angle symbol+30degrees)-4 as a transformation of the function y= sin(angle).
1. The amplitude is 2.
Since time does not appear in the equation, I cannot say what the period is.
The phase shift is pi/2
The vertical shift is -5
2. The function y= -5sin4 (angle symbol+30degrees)-4
has five times the amplitude of y= sin(angle).
It is also shifted vertically by -4, and has a phase shoft of -30 degrees.
Allow me to make a small correction to drwls solution.
in 1. the equation was given as
y= 2sin[4(angle symbol)-pie/2]-5
to read the phase shift directly from the equation the factor of 4 should be outside the bracket, so change it to
y= 2sin4[(angle symbol)-pie/8]-5
and now the phase shift is pi/8 to the right.
The period is 2pi/k for y = asink(theta)
so here the period is 2pi/4 or pi/2
1. To state the amplitude, period, phase shift, and vertical shift for the function y = 2sin[4(angle symbol)-pi/2]-5, let's break down the different parts:
Amplitude: The amplitude is the absolute value of the coefficient in front of the sine function. In this case, the coefficient is 2, so the amplitude is 2.
Period: The period can be found by dividing 2π by the coefficient of the angle symbol. Here, the coefficient is 4, so the period is 2π/4 = π/2.
Phase Shift: The phase shift can be determined by finding the value inside the brackets (angle symbol - pi/2). To shift the graph horizontally, we solve for angle symbol when the expression inside the brackets equals zero. angle symbol - pi/2 = 0, so angle symbol = pi/2. This means that the graph is shifted to the right by pi/2.
Vertical Shift: The vertical shift is given by the constant outside the sine function, which is -5 in this case. The graph is shifted downward by 5 units.
Therefore, the function y = 2sin[4(angle symbol)-pi/2]-5 has an amplitude of 2, a period of pi/2, a phase shift to the right by pi/2, and a vertical shift downward by 5 units.
2. To describe the function y = -5sin4 (angle symbol+30degrees)-4 as a transformation of the function y = sin(angle), we can identify the changes made to the original function:
Reflection: The negative sign in front of the sine function, -5sin, indicates that the graph is reflected over the x-axis. This means that the values of y will be negative compared to the original sine function.
Amplitude: The coefficient in front of the sine function, -5, represents the amplitude. It indicates that the amplitude of the graph is multiplied by 5 compared to the original function. So the amplitude is 5.
Frequency: The coefficient of the angle symbol, 4, determines the frequency of the graph. Since it is greater than 1, the graph will have more oscillations within the same interval as the original function. The period is calculated by dividing 2π by the coefficient, so the period is 2π/4 = π/2.
Phase Shift: The angle symbol is shifted by +30 degrees, which means the graph is shifted to the left by 30 degrees.
Vertical Shift: The constant term, -4, represents the vertical shift. The graph is shifted downward by 4 units compared to the original function.
In summary, the function y = -5sin4 (angle symbol+30degrees)-4 is a reflection of the original sine function across the x-axis, with an amplitude of 5, a frequency of 4 (more oscillations within the same interval), a phase shift to the left by 30 degrees, and a vertical shift downward by 4 units.