Answer: False
Step-by-step explanation:
The base is more than 1, so the function is increasing.
30 of the divisors are multiples of 3
The factors of 2160 are 1, 2160, 2, 1080, 3, 720, 4, 540, 5, 432, 6, 360, 8, 270, 9, 240, 10, 216, 12, 180, 15, 154, 16, 135, 18, 120, 20, 108, 24, 90, 27, 80, 30, 72, 36, 60, 40, 54, 45, 48
These are all divisors of 2160 but only 30 are multiples of 3.
The ones that are NOT are 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80
You can also do this using prime factors 2x2x2x2x3x3x3x5 = 2160 but it is harder to explain.
Answers:
- Midline: y = 0
- Amplitude = 1.5
- Function: g(x) = 1.5sin(0.5x + pi/4)
There are infinitely many possible ways to answer the third part.
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Explanation:
The midline is the horizontal line that goes through the center of the sine curve. Visually we can see that is y = 0. Another way we can see this is to note how y = 0 is the midpoint of y = 1.5 and y = -1.5, which are the max and min respectively.
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The amplitude is the vertical distance from y = 0 to y = 1.5, and it's also the vertical distance from y = 0 to y = -1.5; in short, it's the vertical distance from center to either the peak or valley.
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The general format of a sine curve is
y = A*sin(B(x-C)) + D
where,
- |A| = amplitude
- B = variable tied with the period T, specifically B = 2pi/T
- C = handles the left and right phase shift
- D = handles the vertical shifting, and y = D is the midline.
In this case, we found so far that
- |A| = 1.5 which could lead to A = 1.5
- B = 2pi/(4pi) = 0.5 since T = 4pi is the period
- D = 0
The only thing we're missing is the value of C, which is the phase shift.
Note the point (pi/2, 1.5) is one of the max points on this curve. Also, recall that sin(x) maxes out at 1 when x = pi/2
This must mean that the stuff inside the sine, the B(x-C) portion, must be equal to pi/2 in order to lead sin(B(x-C)) = 1.
So,
B(x-C) = pi/2
0.5(pi/2-C) = pi/2
pi/4 - C/2 = pi/2
4*( pi/4 - C/2 ) = 4*(pi/2)
pi - 2C = 2pi
-2C = 2pi - pi
-2C = pi
C = pi/(-2)
C = -pi/2
This allows us to update the function to get g(x) = 1.5*sin(0.5(x+pi/2)) which is the same as g(x) = 1.5sin(0.5x + pi/4)
This is one possible answer because we could have infinitely many possible values for C, due to sin(x) = 1 having infinitely many solutions.
Also, you could use cosine instead of sine. Cosine is just a phase shifted version and of sine, and vice versa.
It’s acute, equilateral and isosceles
