Question

The above rollercoaster graph needs a function to go with it. Select the correct values that make

the model begin at (0,12) and end at (12,0) with a local minimum at (5,1) and a local maximum at
(9,6).
P+Qcos (2)
f(z)= -2.5 cos(R(z-5))+3.5
0≤z<5
5≤z<9
3-3 sin((z-5)) 9≤z≤ 12.
P [Select]
Q [Select]
R=[Select]
S
[Select]

Answer 1

Explanation:

The function that satisfies the given criteria can be constructed by combining three different equations to form three segments that fit together seamlessly. The general form of the function is:

f(z) = {P + Qcos(R(z-S))} + T

where P, Q, R, S, and T are constants to be determined.

The function has a local minimum at (5,1) and a local maximum at (9,6), which means that the middle segment of the function should be a cosine function that starts at a maximum and ends at a minimum. This can be achieved by setting the middle segment to:

P = 5.5 (the average of the local maximum and minimum)

Q = 5/2 (half the difference between the maximum and minimum)

R = π/2 (to make the function a cosine)

S = 7 (the midpoint between the two endpoints of the segment)

T = 0 (since this segment should start and end at 0)

Therefore, the function for the middle segment is:

f(z) = 5.5 + (5/2)cos(π/2(z-7))

The left and right segments of the function should be linear functions that connect the endpoints to the middle segment. To make the function start at (0,12), we can set the left segment to:

P = 12

Q = 0 (since we don't want any oscillations in this segment)

R = 0 (since this segment is a straight line)

S = 0 (since we want the function to start at 0)

T = 0 (since this segment should start at 12)

Therefore, the function for the left segment is:

f(z) = 12 - (12/5)z

To make the function end at (12,0), we can set the right segment to:

P = 0

Q = 0

R = 0

S = 12 (since we want the function to end at 0)

T = -3 (since this segment should end at 0 and we want the middle segment to start at 5.5)

Therefore, the function for the right segment is:

f(z) = -3z + 36

Putting all three segments together, we get:

f(z) = {12 - (12/5)z} 0 ≤ z < 5

f(z) = 5.5 + (5/2)cos(π/2(z-7)) 5 ≤ z < 9

f(z) = -3z + 36 9 ≤ z ≤ 12

Therefore, the values of P, Q, R, S, and T are:

P = 12

Q = 5/2

R = π/2

S = 7

T = 0 (for the middle segment)

T = 12 (for the left segment)

T = -3 (for the right segment)

Therefore, the answer is:

P = 12

Q = 5/2

R = π/2

S = 7

T = 0 (for the middle segment)

T = 12 (for the left segment)

T = -3 (for the right segment)

Answer 2

(P, Q, R, S) = (6.5, 5.5, π/4, -1.5)

Explanation:

You want the values of the parameters P, Q, R, and S that makes the piecewise-defined function match the given graph. The function is ...

`f(x)=\begin{cases}P+Q\cos{\left((\pi)/(5)x\right)}&0\le x < 5\\-2.5cos((R(x-5)))+3.5&5\le x < 9\\3-3\sin{\left((\pi)/(3)(x-S)\right)}&9\le x \le12 \end{cases}`

Amplitude

The value of Q in the function is half the difference between the maximum and minimum on the interval [0, 5). It is ...

Q = (12 -1)/2 = 5.5

Offset

The value of P in the function is the average of the maximum and minimum on the interval [0, 5). It is ...

P = (12 +1)/2 = 6.5

Frequency

The value of R in the function is π divided by the difference between the interval ends. The interval applicable to R is [5, 9). R is ...

R = π/(9 -5) = π/4

Horizontal shift

The value of S in the function is the average of the interval ends. It can be reduced by any multiple of twice the length of the interval. (The reason for that reduction would be to make the number have as small a magnitude as possible.) The interval applicable to S is [9, 12]. S is ...

S = (9 +12)/2 -n·(2(12-9)) = 10.5 -6n

For n = 2, the value of S is ...

S = 10.5 -12 = -1.5

Other possible values include 4.5 and 10.5. Your answer checker may have a preference for one or another of these values.

The parameters in the function are ...

(P, Q, R, S) = (6.5, 5.5, π/4, -1.5)

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Additional comment

Our description of the parameters in terms of the interval ends is based on the fact that each interval includes exactly 1/2 period of the trig function. For the sine function, the horizontal shift is based on the negative-going midline crossing, halfway between the extremes at the interval ends.

The attached graph provides confirmation that our choice of parameters is appropriate.