Phase Shift Basics: Identify Horizontal Shifts In Trig Functions

1) What horizontal shift transforms y = sin x into y = sin(x + 7π/6)?

Right 7π/6

Left 7π/6

Right 7π/12

Left 7π/12

y = sin(x + a) shifts left by a.

So, the final answer is left 7π/6.

Explanation

y = sin(x + a) shifts left by a.

So, the final answer is left 7π/6.

2) Which function is obtained by shifting y = −sin x to the right by π/10?

Y = −sin(x − π/10)

Y = −sin(x + π/10)

Y = sin(x − π/10)

Y = sin(x + π/10)

Right shift by a: replace x with (x − a); keep the leading minus.

So, the final answer is y = −sin(x − π/10).

Explanation

Right shift by a: replace x with (x − a); keep the leading minus.

So, the final answer is y = −sin(x − π/10).

3) Consider y = cos(½x + π/6). What is the phase shift?

Left π/6

Right π/6

Left π/3

Right π/3

Shift = left by φ/b with φ = π/6, b = 1/2 ⇒ (π/6)/(1/2) = π/3 left.

So, the final answer is left π/3.

Explanation

Shift = left by φ/b with φ = π/6, b = 1/2 ⇒ (π/6)/(1/2) = π/3 left.

So, the final answer is left π/3.

4) Two functions are given: f(x) = sin x and g(x) = sin(x − 2π/3). Which describes g relative to f?

G is f shifted left 2π/3

G is f shifted right 2π/3

G is f shifted right π/3

G is f shifted left π/3

(x − a) shifts right by a.

So, the final answer is “shifted right 2π/3”.

Explanation

(x − a) shifts right by a.

So, the final answer is “shifted right 2π/3”.

5) Which expression represents a left shift of π/8 for y = cos(5x)?

Y = cos(5x − 5π/8)

Y = cos(5x + 5π/8)

Y = cos(5x + π/8)

Y = cos(5x − π/8)

Left shift by a: cos[5(x + a)] = cos(5x + 5a); a = π/8 ⇒ +5π/8.

So, the final answer is y = cos(5x + 5π/8).

Explanation

Left shift by a: cos[5(x + a)] = cos(5x + 5a); a = π/8 ⇒ +5π/8.

So, the final answer is y = cos(5x + 5π/8).

6) A sine wave y = sin x is shifted so that its midline crossing going upward occurs at x = π instead of x = 0. What is the new equation?

Y = sin(x − π)

Y = sin(x + π)

Y = sin(2x − π)

Y = sin(½x − π)

The “upward midline crossing” of sin occurs at its zeros; move 0 → π with right shift.

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

Explanation

The “upward midline crossing” of sin occurs at its zeros; move 0 → π with right shift.

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

7) If y = tan(2x − π/2), what is the phase shift when written as y = tan[2(x − h)]?

H = −π/2 (left)

H = π/2 (right)

H = π/4 (right)

H = −π/4 (left)

2x − π/2 = 2(x − π/4) ⇒ h = π/4 (right).

So, the final answer is h = π/4 (right).

Explanation

2x − π/2 = 2(x − π/4) ⇒ h = π/4 (right).

So, the final answer is h = π/4 (right).

8) The function y = −2sin(3(x + π/9)) has which horizontal shift relative to y = sin x?

Left π/9

Right π/9

Left π/3

Right π/3

(x + π/9) is a left shift by π/9; (−2) only reflects/scales vertically.

So, the final answer is left π/9.

Explanation

(x + π/9) is a left shift by π/9; (−2) only reflects/scales vertically.

So, the final answer is left π/9.

9) Which function is y = cos(x − π/3) equivalent to in terms of horizontal translation of y = cos x?

Shift left π/3

Shift right π/3

Shift right π/6

Shift left π/6

(x − a) shifts right by a.

So, the final answer is shift right π/3.

Explanation

(x − a) shifts right by a.

So, the final answer is shift right π/3.

10) Write y = sin(4x + π) in the form y = sin[4(x − h)]. What is h?

H = −π/4 (left)

H = π/4 (right)

H = −π (left)

H = π (right)

4x + π = 4(x + π/4) = 4[x − (−π/4)] ⇒ h = −π/4.

So, the final answer is h = −π/4 (left).

Explanation

4x + π = 4(x + π/4) = 4[x − (−π/4)] ⇒ h = −π/4.

So, the final answer is h = −π/4 (left).

11) What is the phase shift of y = sin(x − π/3)?

π/3 to the left

π/3 to the right

π/6 to the right

π/6 to the left

Inside (x − π/3) means shift right by π/3.

So, the final answer is π/3 to the right.

Explanation

Inside (x − π/3) means shift right by π/3.

So, the final answer is π/3 to the right.

12) The graph of y = cos x is shifted left by π/5 and reflected across the x-axis. Which equation represents the result?

Y = −cos(x − π/5)

Y = cos(−x − π/5)

Y = −cos(x + π/5)

Y = cos(x + π/5)

Left shift: cos(x + π/5); reflection: add a negative outside.

So, the final answer is y = −cos(x + π/5).

Explanation

Left shift: cos(x + π/5); reflection: add a negative outside.

So, the final answer is y = −cos(x + π/5).

13) Which of the following has no horizontal shift?

Y = sin(2x)

Y = sin(2x − π)

Y = sin(x + π/2)

Y = sin(x − π)

Multiplying x by 2 changes period, not phase; others have ± constants.

So, the final answer is y = sin(2x).

Explanation

Multiplying x by 2 changes period, not phase; others have ± constants.

So, the final answer is y = sin(2x).

14) If y = cos x is shifted to become y = cos(x + a), and the resulting graph starts at a maximum at x = −π/3 instead of x = 0, what is a?

A = −π/3

A = π/3

A = π/6

A = −π/6

To move the max from 0 to −π/3, shift left π/3 ⇒ use (x + π/3).

So, the final answer is a = π/3.

Explanation

To move the max from 0 to −π/3, shift left π/3 ⇒ use (x + π/3).

So, the final answer is a = π/3.

15) Consider y = sin(3x − 3π/2). What is the phase shift?

Right π/2

Right π/3

Right π/6

Right π/4

Shift = left by φ/b with φ = −3π/2, b = 3 ⇒ left by (−3π/2)/3 = −π/2 ⇒ right π/2.

So, the final answer is right π/2.

Explanation

Shift = left by φ/b with φ = −3π/2, b = 3 ⇒ left by (−3π/2)/3 = −π/2 ⇒ right π/2.

So, the final answer is right π/2.

16) Which function is equivalent to shifting y = tan x right by π/6?

Y = tan(x + π/6)

Y = tan(2x − π/6)

Y = tan(½x − π/6)

Y = tan(x − π/6)

Right shift by a means replace x with (x − a).

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

Explanation

Right shift by a means replace x with (x − a).

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

17) Identify the horizontal shift for y = 3sin(½x + π/4).

Left π/4

Right π/4

Left π/2

Right π/2

For bx + φ, shift = left by φ/b. Here b = 1/2, φ = π/4 ⇒ shift = (π/4)/(1/2) = π/2 left.

So, the final answer is left π/2.

Explanation

For bx + φ, shift = left by φ/b. Here b = 1/2, φ = π/4 ⇒ shift = (π/4)/(1/2) = π/2 left.

So, the final answer is left π/2.

18) The function y = −cos(x − π/2) is the result of which transformation applied to y = cos x?

Reflect across x-axis and shift right π/2

Reflect across y-axis and shift right π/2

Reflect across x-axis and shift left π/2

Shift left π/2 only

(x − π/2) shifts right π/2; the negative sign reflects across x-axis.

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

Explanation

(x − π/2) shifts right π/2; the negative sign reflects across x-axis.

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

19) Which function is obtained by shifting y = sin x to the left by π/4?

Y = sin(x − π/4)

Y = sin(x + π/2)

Y = sin(x − π/2)

Y = sin(x + π/4)

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

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

Explanation

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

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

20) The function y = cos(2x + π) can be written in the form y = cos[2(x + c)]. What is c?

C = π/2 to the left

C = π/2 to the right

C = −π/2 to the right

C = π to the right

2x + π = 2(x + π/2), so c = π/2 (left shift).

So, the final answer is c = π/2 to the left.

Explanation

2x + π = 2(x + π/2), so c = π/2 (left shift).

So, the final answer is c = π/2 to the left.

Phase Shift Basics: Identify Horizontal Shifts in Trig Functions

1) What horizontal shift transforms y = sin x into y = sin(x + 7π/6)?

2)

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

2) Which function is obtained by shifting y = −sin x to the right by π/10?

3) Consider y = cos(½x + π/6). What is the phase shift?

4) Two functions are given: f(x) = sin x and g(x) = sin(x − 2π/3). Which describes g relative to f?

5) Which expression represents a left shift of π/8 for y = cos(5x)?

6) A sine wave y = sin x is shifted so that its midline crossing going upward occurs at x = π instead of x = 0. What is the new equation?

7) If y = tan(2x − π/2), what is the phase shift when written as y = tan[2(x − h)]?

8) The function y = −2sin(3(x + π/9)) has which horizontal shift relative to y = sin x?

9) Which function is y = cos(x − π/3) equivalent to in terms of horizontal translation of y = cos x?

10) Write y = sin(4x + π) in the form y = sin[4(x − h)]. What is h?

11) What is the phase shift of y = sin(x − π/3)?

12) The graph of y = cos x is shifted left by π/5 and reflected across the x-axis. Which equation represents the result?

13) Which of the following has no horizontal shift?

14) If y = cos x is shifted to become y = cos(x + a), and the resulting graph starts at a maximum at x = −π/3 instead of x = 0, what is a?

15) Consider y = sin(3x − 3π/2). What is the phase shift?

16) Which function is equivalent to shifting y = tan x right by π/6?

17) Identify the horizontal shift for y = 3sin(½x + π/4).

18) The function y = −cos(x − π/2) is the result of which transformation applied to y = cos x?

19) Which function is obtained by shifting y = sin x to the left by π/4?

20) The function y = cos(2x + π) can be written in the form y = cos[2(x + c)]. What is c?