Modeling With Phase Shifts: Real-World Trig Translations

1) Which function represents a cosine graph shifted π/3 units to the right?

Y = cos(x + π/3)

Y = cos(x − π/3)

Y = cos(x) + π/3

Y = cos(x) − π/3

A right shift by π/3 means replacing x with (x − π/3).

So, the final answer is y = cos(x − π/3).

Explanation

A right shift by π/3 means replacing x with (x − π/3).

So, the final answer is y = cos(x − π/3).

2) The function y = sin(x + π/2) is equivalent to which basic trig function with no horizontal shift?

Y = cos(x)

Y = −cos(x)

Y = sin(x)

Y = −sin(x)

Identity — sin(x + π/2) = cos(x).

So, the final answer is y = cos(x).

Explanation

Identity — sin(x + π/2) = cos(x).

So, the final answer is y = cos(x).

3) Consider y = 2sin(3(x − π/6)) + 1. What is the horizontal (phase) shift?

Left π/6

Right π/6

Left π/3

Right π/3

(x − π/6) indicates a right shift by π/6.

So, the final answer is right π/6.

Explanation

(x − π/6) indicates a right shift by π/6.

So, the final answer is right π/6.

4) Which of the following is a phase shift of the parent tangent function to the left by π/4?

Y = tan(x − π/4)

Y = tan(2x + π/4)

Y = tan(2x − π/4)

Y = tan(x + π/4)

Left shift by a ⇒ (x + a).

So, the final answer is y = tan(x + π/4).

Explanation

Left shift by a ⇒ (x + a).

So, the final answer is y = tan(x + π/4).

5) The function y = −cos(x − π/2) has been transformed from y = cos(x) by:

Left shift π/2 only

Right shift π only

Right shift π/2 and reflection across x-axis

Left shift π/2 and reflection across y-axis

Step 1: (x − π/2) → right shift π/2.

Step 2: The negative sign → reflection over the x-axis.

So, the final answer is right shift π/2 and reflection across x-axis.

Explanation

Step 1: (x − π/2) → right shift π/2.

Step 2: The negative sign → reflection over the x-axis.

So, the final answer is right shift π/2 and reflection across x-axis.

6) A sinusoid has midline y = 0, amplitude 1, period 2π, and passes through (x, y) = (π/2, 1) as a maximum. Which equation matches?

Y = sin(x − π/2)

Y = cos(x − π/2)

Y = cos(x + π/2)

Y = sin(x + π/2)

Step 1: The cosine curve reaches a maximum where x = phase shift.

Step 2: Since max is at π/2, use y = cos(x − π/2).

So, the final answer is y = cos(x − π/2).

Explanation

Step 1: The cosine curve reaches a maximum where x = phase shift.

Step 2: Since max is at π/2, use y = cos(x − π/2).

So, the final answer is y = cos(x − π/2).

7) A graph described as "the basic sine wave shifted left by π" can be written as:

Y = sin(x + π)

Y = sin(x − π)

Y = −sin(x − π/2)

Y = −sin(x + π/2)

Left shift by a means y = sin(x + a).

So, the final answer is y = sin(x + π).

Explanation

Left shift by a means y = sin(x + a).

So, the final answer is y = sin(x + π).

8) A cosine graph with amplitude 2, period 2π, and phase shift right π/3 is:

Y = 2cos(x + π/3)

Y = 2cos(x − π/3)

Y = 2cos(2x − π/3)

Y = 2cos(2x + π/3)

Step 1: Right shift π/3 → (x − π/3).

Step 2: Amplitude 2 → multiply by 2.

So, the final answer is y = 2cos(x − π/3).

Explanation

Step 1: Right shift π/3 → (x − π/3).

Step 2: Amplitude 2 → multiply by 2.

So, the final answer is y = 2cos(x − π/3).

9) To move y = sin(x) so that its zero at x = 0 becomes a zero at x = −π/3 (with identical shape and period), which function works?

Y = sin(x − π/3)

Y = sin(x + π/3)

Y = sin(x − 2π/3)

Y = sin(x + 2π/3)

Step 1: New zero at −π/3 → shift left π/3.

Step 2: Left shift → (x + π/3).

So, the final answer is y = sin(x + π/3).

Explanation

Step 1: New zero at −π/3 → shift left π/3.

Step 2: Left shift → (x + π/3).

So, the final answer is y = sin(x + π/3).

10) The function y = 4sin(½x − π/8) can be written y = 4sin(½(x − c)). What is c and the horizontal shift direction?

C = π/8; shift right

C = π/8; shift left

C = π/4; shift right

C = π/4; shift left

Step 1: ½x − π/8 = ½(x − π/4).

Step 2: Therefore, c = π/4 (right shift).

So, the final answer is c = π/4; shift right.

Explanation

Step 1: ½x − π/8 = ½(x − π/4).

Step 2: Therefore, c = π/4 (right shift).

So, the final answer is c = π/4; shift right.

11) The graph of y = cos(x) is shifted to become one with its maximum at x = −π instead of x = 0. Which equation matches this shift?

Y = cos(x − π)

Y = cos(x) + π

Y = cos(x + π)

Y = cos(x − 2π)

Moving the max from 0 to −π means shifting left by π → (x + π).

So, the final answer is y = cos(x + π).

Explanation

Moving the max from 0 to −π means shifting left by π → (x + π).

So, the final answer is y = cos(x + π).

12) Rewrite y = sin(2x − π) in the form y = sin(2(x − c)). What is c?

π

π/2

−π/2

−π

Step 1: 2x − π = 2(x − π/2).

Step 2: Therefore, c = π/2 (right shift).

So, the final answer is c = π/2.

Explanation

Step 1: 2x − π = 2(x − π/2).

Step 2: Therefore, c = π/2 (right shift).

So, the final answer is c = π/2.

13) Consider y = 3cos(4x + π). What is the phase shift, written y = 3cos(4(x − c))?

C = −π

C = −π/4

C = π/4

C = π

Step 1: 4x + π = 4(x + π/4) = 4(x − (−π/4)).

Step 2: Hence c = −π/4 (left shift).

So, the final answer is c = −π/4.

Explanation

Step 1: 4x + π = 4(x + π/4) = 4(x − (−π/4)).

Step 2: Hence c = −π/4 (left shift).

So, the final answer is c = −π/4.

14) Which pair of functions are horizontally shifted versions of each other with the same period and amplitude?

Y = sin(x) and y = sin(2x)

Y = cos(x) and y = cos(x) + 1

Y = sin(x) and y = sin(x − π/6)

Y = tan(x) and y = −tan(x)

The second function is a right-shifted version of sin(x).

So, the final answer is y = sin(x) and y = sin(x − π/6).

Explanation

The second function is a right-shifted version of sin(x).

So, the final answer is y = sin(x) and y = sin(x − π/6).

15) The function y = sin(x) is shifted to y = sin(x − a). If the maximum originally at x = π/2 moves to x = 5π/6, what is a?

A = π/6

A = −π/3

A = −π/6

A = π/3

Step 1: Shift amount = new − old = 5π/6 − π/2 = π/3.

Step 2: Since it moved right, a = π/3.

So, the final answer is a = π/3.

Explanation

Step 1: Shift amount = new − old = 5π/6 − π/2 = π/3.

Step 2: Since it moved right, a = π/3.

So, the final answer is a = π/3.

16) A tangent graph has vertical asymptotes now at x = −π/2 and x = 3π/2 (instead of −π/2 and π/2). What horizontal shift occurred?

Right π

Left π

Right π/2

Left π/2

Step 1: The interval between asymptotes is π.

Step 2: The new rightmost asymptote moved π units to the right.

So, the final answer is right π.

Explanation

Step 1: The interval between asymptotes is π.

Step 2: The new rightmost asymptote moved π units to the right.

So, the final answer is right π.

17) Rewrite y = −sin(x + 3π/4) as y = −sin(x − c). What is c?

C = −3π/4

C = 3π/4

C = −π/4

C = π/4

Step 1: (x + 3π/4) = (x − (−3π/4)) ⇒ c = −3π/4, meaning shift left 3π/4.

However, written as y = −sin(x − c), c = 3π/4 for the rightward equivalent.

So, the final answer is c = 3π/4.

Explanation

Step 1: (x + 3π/4) = (x − (−3π/4)) ⇒ c = −3π/4, meaning shift left 3π/4.

However, written as y = −sin(x − c), c = 3π/4 for the rightward equivalent.

So, the final answer is c = 3π/4.

18) A sine curve with period 2π has its first maximum at x = π/2 and crosses the midline going upward at x = 0. Which model best fits?

Y = sin(x − 2)

Y = sin(x − π/2)

Y = sin(x) − 2

Y = sin(x)

The base sine function already crosses at 0 and peaks at π/2.

So, the final answer is y = sin(x).

Explanation

The base sine function already crosses at 0 and peaks at π/2.

So, the final answer is y = sin(x).

19) Which statement about y = cos(2(x + π/6)) is true?

Shift left π/6; period is π

Shift right π/6; period is 2π

Shift left π/6; period is 2π

No shift; period is π

Step 1: (x + π/6) ⇒ left shift π/6.

Step 2: b = 2 ⇒ period = π.

So, the final answer is shift left π/6; period = π.

Explanation

Step 1: (x + π/6) ⇒ left shift π/6.

Step 2: b = 2 ⇒ period = π.

So, the final answer is shift left π/6; period = π.

20) A cosine function has period 2π, amplitude 1, and now has a minimum at x = 0 (instead of a maximum). Which is correct?

Y = cos(x) − π

Y = −cos(x − π/2)

Y = cos(x − π/2)

Y = −cos(x)

To invert maximum to minimum, multiply by −1.

So, the final answer is y = −cos(x).

Explanation

To invert maximum to minimum, multiply by −1.

So, the final answer is y = −cos(x).

Modeling with Phase Shifts: Real-World Trig Translations

1) Which function represents a cosine graph shifted π/3 units to the right?

2)

What first name or nickname would you like us to use?

2) The function y = sin(x + π/2) is equivalent to which basic trig function with no horizontal shift?

3) Consider y = 2sin(3(x − π/6)) + 1. What is the horizontal (phase) shift?

4) Which of the following is a phase shift of the parent tangent function to the left by π/4?

5) The function y = −cos(x − π/2) has been transformed from y = cos(x) by:

6) A sinusoid has midline y = 0, amplitude 1, period 2π, and passes through (x, y) = (π/2, 1) as a maximum. Which equation matches?

7) A graph described as "the basic sine wave shifted left by π" can be written as:

8) A cosine graph with amplitude 2, period 2π, and phase shift right π/3 is:

9) To move y = sin(x) so that its zero at x = 0 becomes a zero at x = −π/3 (with identical shape and period), which function works?

10) The function y = 4sin(½x − π/8) can be written y = 4sin(½(x − c)). What is c and the horizontal shift direction?

11) The graph of y = cos(x) is shifted to become one with its maximum at x = −π instead of x = 0. Which equation matches this shift?

12) Rewrite y = sin(2x − π) in the form y = sin(2(x − c)). What is c?

13) Consider y = 3cos(4x + π). What is the phase shift, written y = 3cos(4(x − c))?

14) Which pair of functions are horizontally shifted versions of each other with the same period and amplitude?

15) The function y = sin(x) is shifted to y = sin(x − a). If the maximum originally at x = π/2 moves to x = 5π/6, what is a?

16) A tangent graph has vertical asymptotes now at x = −π/2 and x = 3π/2 (instead of −π/2 and π/2). What horizontal shift occurred?

17) Rewrite y = −sin(x + 3π/4) as y = −sin(x − c). What is c?

18) A sine curve with period 2π has its first maximum at x = π/2 and crosses the midline going upward at x = 0. Which model best fits?

19) Which statement about y = cos(2(x + π/6)) is true?

20) A cosine function has period 2π, amplitude 1, and now has a minimum at x = 0 (instead of a maximum). Which is correct?