π Phase shift sine cosine functions (14 MCQs)
π From Calculus β’ 1. Basics before calculus β’ 14 questions available
What is Phase shift sine cosine functions?
Definition:
Phase shift in sine and cosine functions refers to the horizontal translation of the graph, given by or , where is the phase shift (right if , left if ).
Example:
For , rewrite as , so phase shift is units to the right.
Reason:
Phase shift models delays or advancements in waves, like sound phase differences or alternating current circuits.
π All Phase shift sine cosine functions MCQs
Q1. In the function y = A sin(Bx - C), what does the absolute value of A represent?
π Explanation: The magnitude of A determines how far the sine wave moves above and below the horizontal axis. This distance from the midline to the peak (or trough) is called the amplitude. Therefore, |A| directly gives the amplitude of the graph, independent of any horizontal shifts or changes in period.
Q2. If the graph of y = 3 sin(2x - Ο/4) is shifted Ο/4 units to the right, what is the new phase shift C in the standard form y = 3 sin(2x - C)?
π Explanation: Shifting a sinusoid right by Ο/4 adds that amount to the existing horizontal displacement. The original form already contains a subtraction of Ο/4 inside the argument. Adding another Ο/4 to the right results in the combined shift of Ο/4, so the new C equals Ο/4, matching option B.
Q3. Suppose y = 4 cos(5x + C) passes through the point (0,0). What value of C (in radians) must hold?
π Explanation: At x = 0 the cosine term reduces to cos(C). For the product to be zero, cos(C) must be zero, which occurs at C = Ο/2 plus integer multiples of Ο. The smallest positive solution is Ο/2, making option B the correct choice.
Q4. Given y = -2 sin(3x - C) has its maximum at x = Ο/6, determine C.
π Explanation: A maximum of a sine function occurs when its argument equals Ο/2 (plus 2Οk). Set 3(Ο/6) - C = Ο/2, giving Ο/2 - C = Ο/2, so C = 0. However the negative sign flips the graph, turning a maximum into a minimum; thus the original maximum of the unβnegated sine corresponds to C = Ο/6. Therefore the correct value is Ο/6.
Q5. Compare the periods of y = 2 sin(4x) and y = 5 cos(2x). Which statement is true?
π Explanation: The period of a sinusoid is 2Ο divided by the coefficient B. For y = 2 sin(4x), the period is 2Ο/4 = Ο/2. For y = 5 cos(2x), the period is 2Ο/2 = Ο. Hence the first period (Ο/2) is exactly half of the second period (Ο), confirming option B.
Q6. Which transformation results in the graph of y = sin(x) being reflected over the xβaxis and then vertically stretched by factor 3?
π Explanation: Reflecting over the xβaxis changes the sign of the function, giving -sin(x). A subsequent vertical stretch by a factor of 3 multiplies the entire expression by 3, yielding -3 sin(x). This matches option A, while the other choices either omit the reflection or apply the stretch incorrectly.
Q7. If y = A sin(Bx - C) and y = A cos(Bx - C) intersect at x = 0, what relationship must hold among A, B, and C?
π Explanation: At x = 0, the two functions evaluate to A sin(-C) and A cos(-C). For them to be equal, sin(-C) must equal cos(-C). This occurs when -C = Ο/4 + kΟ, the simplest solution being C = Ο/2 (mod Ο). Thus C = Ο/2 ensures equality, which is option C.
Q8. A wheel of radius r rotates with angular speed Ο. The vertical displacement of a point on the rim can be modeled by which function?
π Explanation: When a point on a rotating wheel starts at the top, its vertical coordinate follows a cosine curve shifted downward by Ο radians, representing the opposite side of the circle. The expression rβ―cos(Οtβ―-β―Ο) captures this vertical motion, making option D the appropriate model.
Q9. In modeling daylight hours as D(t) = A cos( (2Ο/365)(t - C) ) + B, what does the parameter C represent?
π Explanation: The term (tβ―-β―C) inside the cosine determines where the peak of the cycle occurs. In the context of daylight, C corresponds to the calendar day on which the maximum daylight length is observed, i.e., the phase shift that aligns the model with the longest day of the year.
Q10. A sound wave is described by y = 0.8 sin(440Ο t + Ο/6). What is the frequency in Hz and the phase shift in degrees?
π Explanation: The angular frequency Ο equals 440Ο, so the ordinary frequency f = Ο/(2Ο) = 440Ο/(2Ο) = 220β―Hz. The phase constant Ο/6 rad converts to (Ο/6)*(180/Ο) = 30Β°. Therefore the wave has a frequency of 220β―Hz and a phase shift of 30Β°, matching option A.
Q11. When converting y = A sin(Bx - C) to an equivalent cosine form y = A cos(Bx - C'), what is the expression for C' in terms of C?
π Explanation: Using the identity sin(ΞΈ) = cos(ΞΈβ―-β―Ο/2), we replace the sine argument with a cosine argument shifted by +Ο/2. Hence, Bxβ―-β―C = (Bxβ―-β―C')β―-β―Ο/2, which leads to C' = C + Ο/2. This relationship is captured by option C.
Q12. If the function y = 2 sin(Οx - Ο/3) is multiplied by -1, which of the following describes the resulting graph?
π Explanation: Multiplying the entire function by -1 flips the graph across the xβaxis, turning all yβvalues into their negatives while preserving shape and amplitude. The horizontal position and period remain unchanged, so the correct description is a reflection over the xβaxis, which is option D.
Q13. Find the values of A, B, and C such that the function y = A sin(Bx - C) passes through (Ο/6,β―1) and (Ο/2,β―-1) and has a period of Ο.
π Explanation: A period of Ο gives B = 2Ο/Ο = 2. Substituting into y = A sin(2x - C) and using the two points leads to the system: sin(Ο/3 - C) = 1/A and sin(Ο - C) = -1/A. Solving yields A = 2 and C = -Ο/6 (equivalently 11Ο/6). Hence option A correctly lists the parameters.
Q14. If the graph of y = 3 cos(2x - C) is shifted upward by 2 units, which of the following represents the new function?
π Explanation: A vertical translation adds a constant to the entire function without altering its internal phase term. Raising the graph by 2 units therefore yields y = 3 cos(2x - C) + 2, which is precisely option A.