The coordinates of the five quarter points for y=sin(x) are given below.
Use these quarter points to determine the y-coordinates of the quarter points of y=2sin(1/4x).
(0,0)
(π/2,1)
(π,0)
(3π/2,−1)
(2π,0)
(Type ordered pairs. Type exact answers, using π as needed. Use integers or fractions for any numbers in the expressions.)
- I asked someone earlier for help, but that person was not specific on how he got the answers. Please do step 1, step 2 e.t.c. This is new to me, so I'll have no idea how to solve questions like this if the steps are broad. I need steps,specifics,work, in order to solve others. I want to know HOW the answers are achieved, not just broad steps. Please teach me. Thank You.
the only thing you need to remember is that sin(kx) has period 2π/k
So, divide that period into four parts, as shown in your example above, then evaluate y at each value of x.
Where do you get stuck in that?
Sure! I can help you understand how to determine the y-coordinates of the quarter points for the equation y = 2sin(1/4x) using the given quarter points for y = sin(x).
Step 1: Understand the equation y = 2sin(1/4x)
In this equation, the 2 in front of sin(1/4x) indicates that the amplitude is doubled compared to the regular sin(x) function. The 1/4 in sin(1/4x) indicates that the period of the function is compressed by a factor of 1/4. It means that you need to divide the x-values of the original quarter points by 1/4 to find the corresponding x-values for the new equation.
Step 2: Calculate the new x-values for the quarter points
To find the new x-values, divide the x-values of the original quarter points by 1/4. The quarter points for y = sin(x) are:
(0, 0), (π/2, 1), (π, 0), (3π/2, -1), (2π, 0)
Dividing the x-values by 1/4 gives:
(0 / (1/4), 0), (π/2 / (1/4), 1), (π / (1/4), 0), (3π/2 / (1/4), -1), (2π / (1/4), 0)
Simplifying the division gives:
(0, 0), (2π, 1), (4π, 0), (6π, -1), (8π, 0)
These are the new x-values for the equation y = 2sin(1/4x).
Step 3: Calculate the new y-coordinates for the quarter points
Now that we have the new x-values, we can substitute them into the equation y = 2sin(1/4x) to find the corresponding y-coordinates.
Using the new x-values, the ordered pairs for the new quarter points are:
(0, 2sin(1/4 * 0)), (2π, 2sin(1/4 * 2π)), (4π, 2sin(1/4 * 4π)), (6π, 2sin(1/4 * 6π)), (8π, 2sin(1/4 * 8π))
Simplifying the expressions gives:
(0, 2sin(0)), (2π, 2sin(1/2π)), (4π, 2sin(π)), (6π, 2sin(3/2π)), (8π, 2sin(2π))
Now, we need to evaluate the sine values for each x-value:
(0, 2 * (0)), (2π, 2 * (1)), (4π, 2 * (0)), (6π, 2 * (-1)), (8π, 2 * (0))
Simplifying gives:
(0, 0), (2π, 2), (4π, 0), (6π, -2), (8π, 0)
These are the y-coordinates for the quarter points of the equation y = 2sin(1/4x).
So, the quarter points for the equation y = 2sin(1/4x) are:
(0, 0), (2π, 2), (4π, 0), (6π, -2), (8π, 0)
I hope that clarifies the steps and process for you! Let me know if you have any other questions.