On the graph of 𝑓(𝑥)=3sin(4𝜋𝑥), 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.

well, you know that two possibilities are P=(3/8,-3) and Q=(5/8,3)

I assume you can find the slope of PQ.

How am I supposed to know that the x-values are 3/8 and 5/8?

To find the slope of the line passing through points P and Q on the graph of f(x) = 3sin(4πx), we need to determine the coordinates of P and Q.

Let's start by finding the x-coordinate of P. The lowest point of the graph occurs when sin(4πx) = -1. Solving for x, we have:

-1 = sin(4πx)
4πx = -π/2 + 2πk (where k is an integer)
x = -1/8 + k/2 (where k is an integer)

Since P occurs before Q, we can choose the value of k such that x is a maximum value less than 0. We choose k = -1, which gives:

x = -1/8 - 1/2 = -5/8

Thus, the x-coordinate of P is -5/8.

Next, let's find the x-coordinate of Q. The highest point of the graph occurs when sin(4πx) = 1. Solving for x, we have:

1 = sin(4πx)
4πx = π/2 + 2πk (where k is an integer)
x = 1/8 + k/2 (where k is an integer)

Again, since P occurs before Q, we choose the value of k such that x is a minimum value greater than -5/8. We choose k = 0, which gives:

x = 1/8

Thus, the x-coordinate of Q is 1/8.

Now we can calculate the slope. The formula for slope is:

slope = (change in y-coordinate) / (change in x-coordinate)

Since the graph is a sine function, which is periodic, the change in y-coordinate from P to Q is the difference between the maximum and minimum values of the function, which is 2.

The change in x-coordinate from P to Q is:

1/8 - (-5/8) = 1/8 + 5/8 = 6/8 = 3/4

Therefore, the slope of the line passing through P and Q is:

slope = (change in y-coordinate) / (change in x-coordinate) = 2 / (3/4) = 8/3

Hence, the slope of the line passing through P and Q is 8/3.