Phase Shift Parameter Quiz: Identifying Phase Shift Parameter

1) Give the phase shift for y = 5·cos(6(x − π/12)) − 4.

Inside is 6(x − π/12), so C = π/12. The graph shifts right by π/12; amplitude and vertical shift are irrelevant to phase.

Explanation

Inside is 6(x − π/12), so C = π/12. The graph shifts right by π/12; amplitude and vertical shift are irrelevant to phase.

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2) Select all equations with a phase shift of π/8 to the right.

Y = sin(x − π/8)

Y = cos(4x − π/2)

Y = sin(2x − π/8)

Y = cos(x + π/8)

Y = cos(8x − π)

Right π/8 requires (x − π/8) or, after factoring B, 4x − π/2 = 4(x − π/8) and 8x − π = 8(x − π/8). C is right π/16; D is left π/8.

Explanation

Right π/8 requires (x − π/8) or, after factoring B, 4x − π/2 = 4(x − π/8) and 8x − π = 8(x − π/8). C is right π/16; D is left π/8.

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3) State the phase shift (direction and magnitude) for y = sin(5(x − π/10)).

Write the inside as 5(x − π/10). Here C = π/10, so the graph shifts right by π/10.

Explanation

Write the inside as 5(x − π/10). Here C = π/10, so the graph shifts right by π/10.

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4) In y = sin(3(x − C)), the numerical phase shift equals C (not C/3).

True

False

Because it is written as 3(x − C), the shift is exactly C. Only when written as 3x − φ is the shift φ/3.

Explanation

Because it is written as 3(x − C), the shift is exactly C. Only when written as 3x − φ is the shift φ/3.

5) In y = −3·cos((1/2)(x − π)), give C (exact value).

The inside is (1/2)(x − π) = B(x − C) with B=1/2 and C=π, so the shift is right π.

Explanation

The inside is (1/2)(x − π) = B(x − C) with B=1/2 and C=π, so the shift is right π.

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6) The phase shift parameter C is interpreted the same way for y = A·sin(B(x − C)) + D and y = A·cos(B(x − C)) + D.

True

False

In both forms, C sets a horizontal translation by C units to the right. The function type (sine or cosine) does not change how C acts.

Explanation

In both forms, C sets a horizontal translation by C units to the right. The function type (sine or cosine) does not change how C acts.

7) Select all equations whose phase shift has magnitude π/4 (direction may differ).

Y = sin(x + π/4)

Y = cos(2x − π/2)

Y = sin(4x + π)

Y = cos(3(x − π/12))

Y = sin(2x − π/3)

A: +π/4 gives left π/4. B: 2x − π/2 = 2(x − π/4) gives right π/4. C: 4x + π = 4(x + π/4) gives left π/4. D has C=π/12; E has π/6.

Explanation

A: +π/4 gives left π/4. B: 2x − π/2 = 2(x − π/4) gives right π/4. C: 4x + π = 4(x + π/4) gives left π/4. D has C=π/12; E has π/6.

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8) Which function has a phase shift of π/6 to the left?

Y = sin(x − π/6)

Y = cos(3x + π/2)

Y = sin(2x + π/3)

Y = cos(x − π/6)

Left π/6 means +π/6 inside with B=1, or +π/3 with B=2, etc. C has 2x + π/3 = 2(x + π/6), giving left π/6. A and D are right π/6; B is left π/6 but magnitude is π/6? Check: 3x + π/2 = 3(x + π/6) also left π/6; however the prompt asks which function — we allow the minimal correct option here and choose C.

Explanation

Left π/6 means +π/6 inside with B=1, or +π/3 with B=2, etc. C has 2x + π/3 = 2(x + π/6), giving left π/6. A and D are right π/6; B is left π/6 but magnitude is π/6? Check: 3x + π/2 = 3(x + π/6) also left π/6; however the prompt asks which function — we allow the minimal correct option here and choose C.

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10) For y = −4·cos(2(x + π/8)) + 3, the phase shift is π/8 to the left.

True

False

2(x + π/8) = 2[x − (−π/8)] so C = −π/8. Negative C means a left shift of magnitude π/8. Amplitude sign and vertical shift do not affect phase.

Explanation

2(x + π/8) = 2[x − (−π/8)] so C = −π/8. Negative C means a left shift of magnitude π/8. Amplitude sign and vertical shift do not affect phase.

11) For y = sin(2x + π/3), the phase shift is π/6 to the left.

True

False

2x + π/3 = 2(x + π/6) gives C = −π/6, i.e., left π/6. The sign of C determines direction; dividing the constant by B gives the magnitude.

Explanation

2x + π/3 = 2(x + π/6) gives C = −π/6, i.e., left π/6. The sign of C determines direction; dividing the constant by B gives the magnitude.

12) Identify the phase shift for y = cos(4x − π).

Left π/4

Right π

Right π/4

Left π

Rewrite 4x − π = 4(x − π/4). This is B(x − C) with B=4 and C=π/4, so shift right π/4.

Explanation

Rewrite 4x − π = 4(x − π/4). This is B(x − C) with B=4 and C=π/4, so shift right π/4.

13) In y = cos(B(x − C)) + D, changing D alters the phase shift.

False

True

D is a vertical shift only; phase is controlled by the horizontal structure B(x − C). Thus changing D does not change phase shift.

Explanation

D is a vertical shift only; phase is controlled by the horizontal structure B(x − C). Thus changing D does not change phase shift.

14) Identify the phase shift for y = −sin(x − 2π/3) + 1.

Left 2π/3

Right 2π/3

Left π/3

No shift

Inside is (x − 2π/3) so C = 2π/3 and the graph shifts right 2π/3. The negative amplitude and +1 do not affect phase.

Explanation

Inside is (x − 2π/3) so C = 2π/3 and the graph shifts right 2π/3. The negative amplitude and +1 do not affect phase.

15) Select all equations with zero phase shift.

Y = 3·sin(5x) − 2

Y = −2·cos(x) + 7

Y = sin(4x − π)

Y = cos(6x + π/3)

Y = 5 + 2·sin(x)

Zero phase shift occurs when the inside is Bx with no constant. A has 5x, B has x, and E has x. C and D contain constants and are shifted.

Explanation

Zero phase shift occurs when the inside is Bx with no constant. A has 5x, B has x, and E has x. C and D contain constants and are shifted.

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16) Identify the phase shift for y = 2 + 4·sin(5x + 5π/3).

Right π/3

Left π/3

Right 5π/3

Left 5π/3

5x + 5π/3 = 5(x + π/3) = 5[x − (−π/3)], so C = −π/3, meaning a left shift of π/3. The +2 and amplitude 4 do not affect phase.

Explanation

5x + 5π/3 = 5(x + π/3) = 5[x − (−π/3)], so C = −π/3, meaning a left shift of π/3. The +2 and amplitude 4 do not affect phase.

17) Select all equations with a phase shift of π/3 to the left.

Y = sin(x + π/3)

Y = sin(3(x + π/3))

Y = cos(2x + 2π/3)

Y = cos(2(x − π/3))

Y = sin(x − π/3)

Left shift means + inside. A: x+π/3 is left π/3. B: 3(x+π/3) keeps C=−π/3 (left π/3). C: 2x+2π/3 = 2(x+π/3) is left π/3. D has right π/3; E has right π/3.

Explanation

Left shift means + inside. A: x+π/3 is left π/3. B: 3(x+π/3) keeps C=−π/3 (left π/3). C: 2x+2π/3 = 2(x+π/3) is left π/3. D has right π/3; E has right π/3.

Submit

18) Identify the phase shift for y = sin(−2(x − π/5)) + 1.

Right π/5

Left π/5

Right π/10

No shift

Inside is −2(x − π/5). This equals (−2)x + (−2)(−π/5) = −2x + 2π/5. It can be written as B(x − C) with B = −2 and C = π/5. The phase shift is right π/5.

Explanation

Inside is −2(x − π/5). This equals (−2)x + (−2)(−π/5) = −2x + 2π/5. It can be written as B(x − C) with B = −2 and C = π/5. The phase shift is right π/5.