On the graph of f(x)=7sin(7pi x), 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
f(x)=7sin(7pi x) has a maginitude of 7
so the largest value is 7 and the minimum value is -7
The period is 2π/ (7π) = 2/7
so P and Q are 1/7 apart along the x-axis
for a period of 2/7 or 4/14
there is a zeros when x = 0
there is a max when x = 1/14
another zero when x = 2/14
there is a min when x = 3/14
and another zero at 4/14
so P is (1/14 , 7) and Q is (3/14, -7)
(check with your calculator, setting your mode to radians, and subbing in x = 1/14 and 3/14 )
slope PQ = (-7-7)/(3/14 - 1/14) = -14/(1/7)
= -14(7) = -98
To find the slope of the line passing through points P and Q, we need to determine the coordinates of P and Q first.
The function f(x) = 7sin(7πx) represents a sinusoidal wave that repeats itself every 2π/7 units in the x-direction. The lowest and highest points occur at the zeroes and peak values of the function, respectively.
At the lowest point, the value of sin(7πx) is -1 since the sine function reaches its lowest value at x = 0. Therefore, we can set f(x) equal to -7 to find the corresponding x-coordinate.
-7 = 7sin(7πx)
Dividing by 7, we get:
-1 = sin(7πx)
Using inverse sine (arcsine) on both sides, we find:
7πx = arcsin(-1)
Since arcsin(-1) equals -π/2, the equation becomes:
7πx = -π/2
Simplifying, we find:
x = -1/14
So, the x-coordinate of point P is -1/14.
To find the x-coordinate of point Q, we need to find the next highest point on the graph. The highest point occurs when sin(7πx) equals 1. Therefore, we can set f(x) equal to 7 to find the corresponding x-coordinate.
7 = 7sin(7πx)
Dividing by 7, we get:
1 = sin(7πx)
Using inverse sine (arcsine) on both sides, we find:
7πx = arcsin(1)
Since arcsin(1) equals π/2, the equation becomes:
7πx = π/2
Simplifying, we find:
x = 1/14
So, the x-coordinate of point Q is 1/14.
Now that we have the coordinates of points P and Q (P: (-1/14, -7) and Q: (1/14, 7)), we can find the slope of the line passing through these points using the formula:
slope = (y2 - y1) / (x2 - x1)
Substituting the values, we get:
slope = (7 - (-7)) / (1/14 - (-1/14))
= 14 / (1/7)
= 14 * 7
= 98
Therefore, the slope of the line passing through points P and Q is 98.
To find the slope of the line passing through points P and Q on the graph of f(x), we need to determine the coordinates of these points first.
The function f(x) = 7sin(7πx) represents a periodic wave, where the graph oscillates between its highest and lowest points. The period of a sine function is 2π divided by the coefficient of x inside the sine function. In this case, the coefficient is 7π, so the period is 2π / (7π) = 2/7.
Since P and Q are consecutive lowest and highest points, we can infer that P is the lowest point and Q is the highest point that occurs right after P. In other words, P is at the start of a wave and Q is at the peak of the next wave.
To find the x-coordinate of P, we need to determine at which point the wave reaches its lowest value. The lowest value of a sine function is -1, so we can set f(x) = -1 and solve for x:
-1 = 7sin(7πx)
Dividing by 7, we get:
-1/7 = sin(7πx)
Using the inverse sine function, we find:
7πx = arcsin(-1/7)
Simplifying further:
x = arcsin(-1/7) / (7π)
This gives us the x-coordinate of P. To find the y-coordinate of P, we substitute the x-coordinate into the original function:
f(x) = 7sin(7πx)
f(P) = 7sin(7π * (arcsin(-1/7) / (7π)))
Simplifying further:
f(P) = 7sin(arcsin(-1/7))
Since sin(arcsin(x)) = x, we find:
f(P) = 7 * (-1/7)
f(P) = -1
Therefore, the coordinates of point P are (arcsin(-1/7) / (7π), -1).
To find the x-coordinate of Q, we need to determine at which point the wave reaches its highest value. The highest value of a sine function is 1, so we can set f(x) = 1 and solve for x:
1 = 7sin(7πx)
Dividing by 7, we get:
1/7 = sin(7πx)
Using the inverse sine function, we find:
7πx = arcsin(1/7)
Simplifying further:
x = arcsin(1/7) / (7π)
This gives us the x-coordinate of Q. To find the y-coordinate of Q, we substitute the x-coordinate into the original function:
f(x) = 7sin(7πx)
f(Q) = 7sin(7π * (arcsin(1/7) / (7π)))
Simplifying further:
f(Q) = 7sin(arcsin(1/7))
Since sin(arcsin(x)) = x, we find:
f(Q) = 7 * (1/7)
f(Q) = 1
Therefore, the coordinates of point Q are (arcsin(1/7) / (7π), 1).
Now that we have the coordinates of points P and Q, we can find the slope of the line passing through them using the formula:
slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1)
Substituting the values, the slope of the line passing through P and Q is:
slope = (1 - (-1)) / (arcsin(1/7) / (7π) - arcsin(-1/7) / (7π))