On the graph of f(x) = 8*sin(3pi*x), points P and Q are at consecutive lowest and highest points with P occurring before Q. Find the slope of the line which passes through P and Q.
Slope = ?
I have tried 48/pi and it didn't work
P is where 3pi*x = 3pi/2
P=(1/2,-8)
Q is where 3pi*x = 5pi/2
Q=(5/6,8)
so, the slope of PQ is 16/(1/3) = 48
To find the slope of the line passing through points P and Q on the graph of f(x) = 8*sin(3pi*x), we need to determine the coordinates of P and Q.
The function f(x) = 8*sin(3pi*x) represents an oscillating curve with a period of 2/3 (since the coefficient of x is 3pi, the period is 2pi divided by 3pi, which simplifies to 2/3). This means that the curve repeats itself every 2/3 units horizontally.
To find the coordinates of P and Q, we need to locate the consecutive lowest and highest points of the curve. The lowest point occurs at x = 1/6, and the highest point occurs at x = 5/6.
For P, substitute x = 1/6 into the function:
f(1/6) = 8*sin(3pi*(1/6))
= 8*sin(pi/2)
= 8*1
= 8
So, P has coordinates (1/6, 8).
For Q, substitute x = 5/6 into the function:
f(5/6) = 8*sin(3pi*(5/6))
= 8*sin(5pi/2)
= 8*(-1)
= -8
So, Q has coordinates (5/6, -8).
Now that we have the coordinates of P and Q, we can find the slope of the line passing through these points using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates:
m = (-8 - 8) / (5/6 - 1/6)
= -16 / 4/6
= -16 * 6/4
= -24
Therefore, the slope of the line passing through points P and Q is -24.
To find the slope of the line passing through points P and Q on the graph of the function f(x) = 8*sin(3πx), we need to determine the coordinates of P and Q.
First, let's understand the behavior of the function f(x) = 8*sin(3πx). The sine function oscillates between its lowest and highest values. In this case, the amplitude is 8, and the period is 2π/3.
To find the x-coordinate of P, we need to find the value of x that corresponds to the lowest point of the period. Since one period of the function is 2π/3, the lowest point occurs at x = (2π/3)(1/4) = π/6. Therefore, the x-coordinate of P is π/6.
Now, let's find the y-coordinate of P by substituting x = π/6 into the function f(x) = 8*sin(3πx):
f(π/6) = 8*sin(3π(π/6)) = 8*sin(π/2) = 8*1 = 8. So, the y-coordinate of P is 8.
Next, to find the x-coordinate of Q, we need to find the value of x that corresponds to the highest point of the period. The highest point occurs at x = (2π/3)(3/4) = (3π/4). Therefore, the x-coordinate of Q is 3π/4.
Finally, let's find the y-coordinate of Q by substituting x = 3π/4 into the function f(x) = 8*sin(3πx):
f(3π/4) = 8*sin(3π(3π/4)) = 8*sin(9π/4) = 8*-1 = -8. So, the y-coordinate of Q is -8.
Now that we have the coordinates of P and Q, we can find the slope of the line passing through these points using the formula:
Slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1)
Substituting the values:
Slope = (-8 - 8) / (3π/4 - π/6)
Slope = (-16) / (7π/12)
Slope = -192π / 7π
Slope = -192/7
So, the slope of the line passing through P and Q is -192/7.