A cos wave has its first maximum at the point (pie/2, 2) and its first minimum at (5pie/2, -4) what is the equation of the function?
I remember writing 3cos(1/2(x-pie/2)-1 but I'm sure that it's wrong ?
well it ranges from -4 to +2
-4 to 2 = 6
so you are right the amplitude is 3
y = 3 cos (a x - phi)
when x = pi/2 , cos (a x - phi) = 1
so
a pi/2 - phi = 0 because cos 0 = 1
when x = 5 pi/2, cos ( a x- phi) = -1
so
a 5 pi/2 - phi = pi, because cos pi = -1
so we have two equations for a and phi
a pi/2 - phi = 0
a 5 pi/2 - phi = pi
---------------------- subtract
a * -4 pi/2 = -pi
2 a = 1
a = 1/2
phi = 1/2 * pi/2 = pi/4
so I get
3 cos (2x - pi/4)
3 cos (x/2 - pi/4)
That is very close to what you have except for that strange -1 at the end.
yes, you need that -1 at the end
So is my equation correct?
yes, it is correct
except that the Greek letter is pi not pie!!
To find the equation of a cosine function based on its maximum and minimum points, we need to consider a few key components: the amplitude, period, phase shift, and vertical shift.
First, let's determine the amplitude. The amplitude is the absolute value of the maximum value (2) or the minimum value (-4), which in this case is 4.
Next, let's determine the period. The period is the distance between two consecutive maximum or minimum points. In this case, the distance between the first maximum at (π/2, 2) and the first minimum at (5π/2, -4) is 4π.
Now, let's determine the phase shift. The phase shift determines the horizontal shift of the graph. In this case, since the first maximum occurs at (π/2, 2), the general equation will be of the form cos(x - c), where c represents the phase shift. The phase shift will be the x-coordinate of the first maximum, so c = π/2.
Finally, let's determine the vertical shift. The vertical shift is the average of the maximum and minimum points or the midline of the graph. In this case, the midline is (2 - 4) / 2 = -1.
Putting it all together, the equation for the given cosine function is:
f(x) = 4 cos(x - π/2) - 1
Therefore, your earlier attempt at writing 3cos(1/2(x-π/2))-1 is incorrect as it does not match the given data. The correct equation is f(x) = 4 cos(x - π/2) - 1.