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=5sin(-π/3x).
(0,0)
(π/2,1)
(π,0)
(3π/2,-1)
(2π,0)
(Type ordered pairs. Type exact answers, using π as needed)
- Please show steps and answers because I want to learn it.
the period of sin(x) is 2π.
the period of sin(-π/3 x) is 2π/(π/3) = 6
You need to multiply by the amplitude.
You need to reflect in the x-axis, since sin(-x) = -sin(x)
so, the quarter points are
(0,0), (3/2,-5), (3,0), (9/2,5), (6,0)
see the graph at
http://www.wolframalpha.com/input/?i=5sin(-%CF%80%2F3+x)
Y=1/5cos (-5x)
Well, I have to say, these coordinates are having quite a party! Let's see what we can do to find the y-coordinates of the quarter points for y = 5sin(-π/3x).
First, let's find the period of the function y = sin(x). The period of y = sin(x) is 2π. Now, let's think about what happens when we multiply the argument of sin(x) by -π/3.
Since the original period is 2π, dividing the period by -π/3 will give us -6. This means that for every x value that goes up by -6, the function will complete one full cycle.
Now, let's look at the given quarter points of y = sin(x):
(0, 0): This is at the beginning of the cycle, so it will remain the same.
(π/2, 1): This is a quarter of the way through the cycle.
(π, 0): This is halfway through the cycle.
(3π/2, -1): This is three-quarters of the way through the cycle.
(2π, 0): This is the end of the cycle.
To find the y-coordinates of the quarter points for y = 5sin(-π/3x), we need to multiply the y-values of the quarter points from y = sin(x) by 5.
So, the quarter points for y = 5sin(-π/3x) are:
(0, 0)
(π/2, 5) since 1 * 5 = 5
(π, 0)
(3π/2, -5) since -1 * 5 = -5
(2π, 0)
And now, the quarter points for y = 5sin(-π/3x) are rockin' and rollin', just like their friends from y = sin(x)!
To find the y-coordinates of the quarter points of the function y = 5sin(-π/3x), we need to substitute each of the x-coordinates from the given quarter points into the function and evaluate it.
Let's go step by step:
1. For the first quarter point (0,0), substitute x = 0 into the function:
y = 5sin(-π/3 * 0)
= 5sin(0)
= 5 * 0
= 0
So the y-coordinate for the first quarter point is (0,0).
2. For the second quarter point (π/2,1), substitute x = π/2 into the function:
y = 5sin(-π/3 * (π/2))
= 5sin(-π/6)
= 5 * (-1/2)
= -5/2
So the y-coordinate for the second quarter point is (π/2, -5/2).
3. For the third quarter point (π,0), substitute x = π into the function:
y = 5sin(-π/3 * π)
= 5sin(-π/3)
= 5 * (-√3/2)
= -5√3/2
So the y-coordinate for the third quarter point is (π, -5√3/2).
4. For the fourth quarter point (3π/2, -1), substitute x = 3π/2 into the function:
y = 5sin(-π/3 * (3π/2))
= 5sin(-π/2)
= 5 * (-1)
= -5
So the y-coordinate for the fourth quarter point is (3π/2, -5).
5. For the fifth quarter point (2π,0), substitute x = 2π into the function:
y = 5sin(-π/3 * (2π))
= 5sin(-2π/3)
= 5 * (√3/2)
= 5√3/2
So the y-coordinate for the fifth quarter point is (2π, 5√3/2).
To summarize, the y-coordinates of the quarter points for y = 5sin(-π/3x) are:
(0, 0)
(π/2, -5/2)
(π, -5√3/2)
(3π/2, -5)
(2π, 5√3/2)