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.