On the graph of f(x)=2sin(6πx), points P and Q are at consecutive lowest and highest points with P occuring before Q. Find the slope of the line which passes through P and Q.
for P
sin T is -1 when T = (3/4)(2pi)
= 3 pi /2
for Q
sin T then becomes +1 when
sin T = 2 pi + pi/2 = 5 pi/2
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now x for P
6 pi x = 3 pi/2
x = 1/4 and y is -2
now x for Q
6 pi x = 5 pi/2
x = (5/12) and y is +2
slope = 4/(5/12 - 1/4)
= 4 / (5/12 - 3/12)
= 4 / (1/6)
= 24
To find the slope of the line passing through points P and Q on the graph of the function f(x), we first need to determine the x-coordinates of these points.
The function f(x) = 2sin(6πx) is a sinusoidal function with a period of 2π/6 = π/3. This means that the function completes one full period of oscillation every π/3 units.
Since P and Q are consecutive lowest and highest points, the horizontal distance between them is equal to half of the period, which is π/3 * 0.5 = π/6.
To find the x-coordinate of point P, we subtract π/6 from the x-coordinate of any highest point on the graph. Similarly, to find the x-coordinate of point Q, we add π/6 to the x-coordinate of P.
Now, let's determine the x-coordinates of P and Q.
Let's choose 0 as a convenient x-coordinate for one of the highest points on the graph. So, the x-coordinate of P will be -π/6, and the x-coordinate of Q will be -π/6 + π/6 = 0.
Now that we have the x-coordinates of P and Q, we can calculate their corresponding y-coordinates by evaluating the function f(x) at these points.
For point P:
f(-π/6) = 2sin(6π * (-π/6)) = 2sin(-π) = 0
For point Q:
f(0) = 2sin(6π * 0) = 2sin(0) = 0
Since both points P and Q have a y-coordinate of 0, they lie on the x-axis.
Now, we have the coordinates of both P and Q: P(-π/6, 0) and Q(0, 0).
To calculate the slope of the line passing through these two points, we can use the formula: slope = (change in y-coordinates) / (change in x-coordinates).
In this case, the change in y-coordinates is 0 - 0 = 0, and the change in x-coordinates is 0 - (-π/6) = π/6.
Therefore, the slope of the line passing through P and Q is 0 divided by π/6, which is 0.
So, the slope of the line passing through points P and Q is 0.