AnswersAlgebra 2 MP 234 CHSModeling with Periodic Functions

Simplifying Rational Expressions Answers

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On a unit circle, the vertical distance from the x-axis to a point on the perimeter of the circle is twice the horizontal distance from the y-axis to the same point. What is sin?

Question illustration
A
One-fifth
Option A
B
StartFraction StartRoot 5 EndRoot Over 5 EndFraction
Option B
C
StartFraction 2 StartRoot 5 EndRoot Over 5 EndFraction
Option C
D
2
Option D
2
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Which is the graph of ?

Question illustration
A
A smooth curve resembling an arctangent function. The x-axis ranges from -6 to 6, and the y-axis from -p to p. The curve approaches horizontal asymptotes at y = p/2 and y = -p/2, passing through the origin.
Option A
B
A graph of a function with asymptotic behavior. The x-axis ranges from -5 to 5, and the y-axis ranges from -p to p. The curve is located in the fourth quadrant, approaching the y-axis asymptotically as x approaches zero from both positive and negative directions. The curve also approaches zero asymptotically as x moves away from zero in both directions. The function does not intersect either axis.
Option B
C
A graph of a function with the x-axis ranging from -2.5 to 2.5 and the y-axis ranging from -p to p. The curve passes through the origin (0,0) and changes direction at y=0. It intersects the x-axis at two points, approximately at x=-1 and x=1. The curve is symmetric about the origin.
Option C
D
The image shows a mathematical graph plotted on a Cartesian coordinate system. The x-axis ranges from -2 to 2, and the y-axis ranges from -p to p. A smooth, continuous curve intersects the x-axis at approximately 1. The curve extends into both the positive and negative quadrants. The point p is labeled on the positive y-axis and -p on the negative y-axis.
Option D
3

A student is given that point P(a, b) lies on the terminal ray of angle , which is between radians and 2 radians. The student uses the steps below to find cos . Step 1Find the quadrant in which P(a, b) lies:P(a, b) is in Quadrant IV.Step 2Use the point and the Pythagorean theorem to determine the value of r:, but since r must be positive, .Step 3Determine cos ., where a and b are positive.Which of the following explains whether the student is correct?

Question illustration
A
The student made an error in step 3 because a is positive in Quadrant IV; therefore, .
Option A
B
The student made an error in step 3 because .
Option B
C
The student made an error in step 2 because r is negative in Quadrant IV; therefore, .
Option C
D
The student made an error in step 2 because using the Pythagorean theorem gives .
Option D
4

Henry is asked to find the exact value of . His steps are shown below.1. Subtract from as many times as possible: – = 2. Find the reference angle for : – = .3. The cosine value for is .4. The cosine value is positive because is in the first quadrant.Which of the following describes Henry’s errors?

Question illustration
A
The reference angle should be , and the sign of the value should be negative.
Option A
B
The reference angle should be , and the cosine value for is .
Option B
C
The sign of the value should be negative, and the reference angle and quadrant should be determined by subtracting in the first step.
Option C
D
The cosine value for is , and the reference angle and quadrant should be determined by subtracting in the first step.
Option D
5

If a vertical line is dropped from the x-axis to the point (12, –9) in the diagram below, what is the value of sec ?

Question illustration
A
Negative five-thirds
Option A
B
Negative five-fourths
Option B
C
Five-fourths
Option C
D
Five-thirds
Option D
6

Which values for have the same reference angles?

Question illustration
A
, ,
Option A
B
, ,
Option B
C
, ,
Option C
D
, ,
Option D
7

Which of the following is true for f(x) = –2sin(x) – 3?

A
The range of the function is the set of real numbers .
Option A
B
The graph of the function is the graph of f(x) = –2sin(x) shifted 3 units up.
C
The amplitude of the function is 2.
D
The period of the function is .
Option D
8

Which statement is true about the graph of the equation ?

Question illustration
A
There is a horizontal asymptote at .
Option A
B
There is a horizontal asymptote at .
Option B
C
There is a vertical asymptote at .
Option C
D
There is a vertical asymptote at .
Option D
9

Which formula gives the zeros of y = sin(x)?

A
for any positive integer k
Option A
B
for any integer k
Option B
C
for any positive integer k
Option C
D
for any integer k
Option D
10

An angle that shares the same sine value of an angle that measures radians is located where?

Question illustration
A
Quadrant I
B
Quadrant II
C
Quadrant IV
D
along an axis
11

The average daily temperature, t, in degrees Fahrenheit for a city as a function of the month of the year, m, can be modeled by the equation , where m = 0 represents January 1, m = 1 represents February 1, m = 2 represents March 1, and so on. Which equation also models this situation?

Question illustration
A
t = negative 35 sine (StartFraction pi Over 6 EndFraction m) + 55
Option A
B
t = negative 35 sine (StartFraction pi Over 6 EndFraction (m + 3)) + 55
Option B
C
t = 35 sine (StartFraction pi Over 6 EndFraction m) + 55
Option C
D
t = 35 sine (StartFraction pi Over 6 EndFraction (m + 3)) + 55
Option D
12

The height, h, in feet of a ball suspended from a spring as a function of time, t, in seconds can be modeled by the equation . What is the height of the ball at its equilibrium?

Question illustration
A
a feet
B
b feet
C
h feet
D
k feet
13

Which statement accurately describes how adding a number, n, to the function affects its graph?

Question illustration
A
There is a vertical shift of n units.
B
The x-intercepts will shift n units.
C
There is a change in amplitude of n units.
D
The range will change by a factor of 2n units.
14

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 a.m. and 3:30 p.m., with a depth of 3.25 meters, while high tides occur at 7:45 a.m. and 11:15 p.m., with a depth of 8.75 meters. Which of the following equations models d, the depth of the water in meters, as a function of time, t, in hours? Let t = 0 be 12:00 a.m.

A
d = negative 3.75 cosine (StartFraction 4 pi Over 31 EndFraction t) + 5
Option A
B
d = negative 3.75 cosine (StartFraction 4 pi Over 29 EndFraction t) + 5
Option B
C
d = negative 2.75 cosine (StartFraction 4 pi Over 31 EndFraction t) + 6
Option C
D
d = negative 2.75 cosine (StartFraction 4 pi Over 29 EndFraction t) + 6
Option D
15

A student uses the equation to represent the speed, s, in feet per second, of a toy car driving around a circular track having an angle of incline , where . To solve the problem, the student used the given value of to find the value of and then substituted the value of in the equation above to solve for s. What is the approximate value of s, the speed of the car in feet per second?

Question illustration
A
5.3
B
7.5
C
9.2
D
28.3

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