13. What is the equation of a cosine function with amplitude 3, transition point (−1, 1), and period p?


A. y = p cos [3(x − 1)] − 1
B. y = 3 cos [2(x − 1)] + 1
C. y = 3 cos [p (x + 1)] − 1
D. y = 3 cos [2(x + 1)] + 1


16. What is the transition point of y = 100 tan (6x) + 4?

A. (0, −4)
B. (6, −4)
C. (0, 4)
D. (6, 4)

20. cot–1 −0.57735 is approximately

A. −1.05.
B. 2.09.
C. −0.65.
D. 2.62.

13.

amplitude 3: 3cos(...)
period p: 3cos(2pi(...)/p) ...
translate by (-1,1): 3cos(2pi(x+1)/p) + 1
So, it appears to be (D), if by p you mean pi.

16: (0,4): (C)

20: (A)

Your use of transition point appears unusual. In #13 the only possible interpretation, given the answer choices, is to use (-1,1) as coordinate translation. Yet transition points are usually max/min or inflection points. Using that same logic gives me (C) on #16.

To answer these questions, we need to understand the properties and formulas of cosine functions, tangent functions, and inverse cotangent functions.

Question 13:
The general equation for a cosine function is y = A cos[B(x - C)] + D, where A is the amplitude, B is the period, C is the horizontal shift, and D is the vertical shift.

In this question, the given parameters are amplitude = 3, transition point (-1, 1), and period = p.

To find the equation of the cosine function, we can plug in the given values into the general equation:
y = A cos[B(x - C)] + D
y = 3 cos[B(x - (-1))] + 1
y = 3 cos[B(x + 1)] + 1

Now, we need to determine the value of B. The period of a cosine function is given by the formula T = 2π/B. In this question, we know that the period is p. Thus, we have:
p = 2π/B

Solving for B, we get:
B = 2π/p

Plugging this value of B into the equation, we have:
y = 3 cos[(2π/p)(x + 1)] + 1

Therefore, the correct answer is option C: y = 3 cos[p(x + 1)] - 1.

Question 16:
The general equation for a tangent function is y = A tan[B(x - C)] + D, where A is the amplitude, B is the period, C is the horizontal shift, and D is the vertical shift.

In this question, the given equation is y = 100 tan(6x) + 4.

To find the transition point, we need to determine the values of C and D. The transition point is the point on the graph where the slope changes. This occurs when the argument of the tangent function (6x in this case) equals π/2.

So, we have:
6x = π/2

Solving for x, we get:
x = π/12

This tells us that the horizontal shift (C) is π/12.

To find the vertical shift (D), we can substitute the transition point into the given equation:
y = 100 tan(6x) + 4
1 = 100 tan(6(π/12)) + 4

Evaluating this expression, we find:
1 = 100 tan(π/2) + 4

Since tan(π/2) is undefined, we know that the vertical shift is 4.

Therefore, the transition point is (π/12, 4).

Hence, the correct answer is option B: (6, -4).

Question 20:
To find the value of cot^-1 (-0.57735), we need to use the formula for the inverse cotangent function:
cot^-1(x) = arccot(x)

The inverse cotangent function gives us the angle whose cotangent is x.

Using a calculator or table, we can find that:
arccot(-0.57735) = 2.09 (approximately)

Therefore, the correct answer is option B: 2.09 (approximately).