Write an expression for the x-values where the the maximum in the expression occurs:
y=6cos((2pi/14)x)-2
I know that the maximum occurs every 14 units if you start at 0,0. I'm stuck after that.
since you labelled this "pre-calculus" , I suppose we can't use the derivative, which would be the easiest way of doing it
consider: y = 6 cos( (2π/14)x)
the period = 2π/(2π/14) = 14
You had that.
so we have a cosine curve with a max of 6 when x = 0 and when x = 14
but we are dropping the whole curve by 2 units, so the maximum will be 4
so the max occurs at
x = 0 , ±14, ±28, ±42, etc , that is every 14 units
general solution:
max of 4 occurs when x = 14k, where k is an integer.
Thanks! Great explanation.
To find the x-values where the maximum occurs in the given expression, we can start by finding the period of the cosine function.
The period of a cosine function is given by the formula:
period = 2π / b, where b is the coefficient of x in the argument of the cosine function.
In this case, the coefficient of x is (2π/14), so we can calculate the period as follows:
period = 2π / (2π/14)
= 14
Now that we know the period is 14 units, we can determine that the maximum will occur every 14 units.
To find the x-values where the maximum occurs, we need to consider the starting point. In this case, there is a horizontal shift of -π/2 units (or -7 units) due to the -2 in the expression.
So, the x-values where the maximum occurs can be obtained by adding multiples of the period (14) to the horizontal shift (-7). In other words, the x-values will be:
x = -7 + 14n, where n is an integer
Therefore, the expression for the x-values where the maximum occurs is:
x = -7 + 14n, where n is an integer.
To find the x-values where the maximum in the expression occurs, we need to understand the characteristics of a cosine function.
The standard form of a cosine function is y = A*cos(Bx + C) + D, where A is the amplitude, B is the frequency (how many cycles fit in 2π), C is the phase shift, and D is the vertical shift.
In your equation, y = 6cos((2π/14)x) - 2, we can see that:
A = 6 (amplitude)
B = 2π/14 (frequency)
C = 0 (no phase shift)
D = -2 (vertical shift)
The maximum value of a cosine function occurs when the cosine function reaches its maximum value of 1, multiplied by the amplitude. Therefore, to find the x-values where the maximum occurs, we need to find the values of x that make cos((2π/14)x) equal to 1.
We can isolate the cosine function by setting cos((2π/14)x) = 1, which leads to:
1 = cos((2π/14)x)
To solve this equation for x, we can take the inverse cosine (arccos) of both sides:
arccos(1) = arccos(cos((2π/14)x))
The inverse cosine of 1 is 0, so we have:
0 = (2π/14)x
Now, we can solve for x by isolating it:
x = (0 * 14)/(2π)
Simplifying further:
x = 0
Therefore, the x-values where the maximum in the expression occurs is x = 0.