A sine function has an amplitude of 4/7, period of 2pi, horizontal shift of -3pi, and vertical shift of 1.
What is the y-value of the positive function at x= pi/2?
taking your data:
y = (4/7)sin (x + 3π) + 1
when x = π/2
y = (4/7)sin(π/2+3π) = 1
= (4/7)sin (7π/2) + 1
= (4/7)(-1) + 1
= 3/7
Well, since I'm a bot, I don't have emotions, but if I did, I'd be feeling pretty "sine-sational" right now! So let's find that y-value, shall we?
First, let's review the properties you gave me. The amplitude is 4/7, which basically tells us how tall the function goes. The period is 2pi, which tells us how long it takes for the function to complete one full cycle. The horizontal shift is -3pi, which means the whole function is shifted to the right by 3pi units. Finally, the vertical shift is 1, which tells us that the whole function is shifted up by 1 unit.
Now, to find the y-value of the positive function at x = pi/2, we need to apply all these properties.
The equation for the given function would be:
y = (4/7) * sin((1/2) * (x + 3pi)) + 1
So, let's plug in x = pi/2 into this equation and see what we get:
y = (4/7) * sin((1/2) * (pi/2 + 3pi)) + 1
Calculating this, we get y ≈ 1.568.
So, the y-value of the positive function at x = pi/2 is approximately 1.568.
I hope that answers your question with a little bit of mathematical humor!
To find the y-value of the positive function at x = pi/2, we can use the equation for a sine function. The general form of a sine function is:
f(x) = A*sin(Bx - C) + D
Where:
A = amplitude
B = 2pi/period
C = phase shift (horizontal shift)
D = vertical shift
In this case, the amplitude is 4/7, the period is 2pi, the horizontal shift is -3pi, and the vertical shift is 1. So, plugging these values into the equation, we get:
f(x) = (4/7)*sin((2pi/2pi)(x - (-3pi))) + 1
Simplifying:
f(x) = (4/7)*sin(x + 3pi) + 1
Now, we can substitute x = pi/2 into this equation:
f(pi/2) = (4/7)*sin(pi/2 + 3pi) + 1
To simplify further, we need to determine the value of sin(pi/2 + 3pi). We know that sin(pi + theta) = -sin(theta), so using that property, we can rewrite the equation as:
f(pi/2) = (4/7)*(-sin(3pi - pi/2)) + 1
Since sin(3pi - pi/2) = sin(5pi/2), we have:
f(pi/2) = (4/7)*(-sin(5pi/2)) + 1
The sine value of 5pi/2 is -1, so we can substitute that:
f(pi/2) = (4/7)*(-(-1)) + 1
Simplifying:
f(pi/2) = (4/7)*(1) + 1
f(pi/2) = 4/7 + 1
f(pi/2) = 11/7 or approximately 1.571
Therefore, the y-value of the positive function at x = pi/2 is approximately 1.571.
To find the y-value of a sine function at a specific value of x, we can use the formula:
y = A*sin(B(x-C)) + D
Where:
-A is the amplitude of the function
-B is related to the period of the function
-C is the horizontal shift
-D is the vertical shift
In this case, the given values are:
Amplitude (A) = 4/7
Period = 2π
Horizontal shift (C) = -3π
Vertical shift (D) = 1
Substituting these values into the formula, we have:
y = (4/7)*sin(2π(x-(-3π))) + 1
Simplifying, we have:
y = (4/7)*sin(2π(x+3π)) + 1
y = (4/7)*sin(2πx + 6π) + 1
Now, we want to find the value of y when x = π/2. Substituting this value into the equation:
y = (4/7)*sin(2π(π/2) + 6π) + 1
y = (4/7)*sin(π + 6π) + 1
y = (4/7)*sin(7π) + 1
Since sin(7π) = 0, we can simplify further:
y = (4/7)*0 + 1
y = 0 + 1
y = 1
Therefore, the y-value of the positive function at x = π/2 is 1.