From Shifted Graphs to Equations: Phase Shift Mastery

1) Which equation represents the graph of y = sin(x) shifted π/3 units to the right?

Y = sin(x + π/3)

Y = sin(x − π/3)

Y = sin(π/3 − x)

Y = sin(−x + π/3)

Step 1: A right shift by π/3 means replacing x with (x − π/3).

Step 2: So, the new equation is y = sin(x − π/3).

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

Explanation

Step 1: A right shift by π/3 means replacing x with (x − π/3).

Step 2: So, the new equation is y = sin(x − π/3).

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

2) The graph of y = cos(x) is shifted left by π/2. Which equation matches the new graph?

Y = cos(x + π/2)

Y = cos(x − π/2)

Y = cos(π/2 − x)

Y = cos(−x − π/2)

Step 1: A left shift by π/2 means adding π/2 inside the parentheses.

Step 2: Therefore, y = cos(x + π/2).

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

Explanation

Step 1: A left shift by π/2 means adding π/2 inside the parentheses.

Step 2: Therefore, y = cos(x + π/2).

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

3) Which function is equivalent to a horizontal shift of y = sin(x) by π/2 to the left?

Y = sin(x + π/2) = cos(x)

Y = sin(x − π/2) = −cos(x)

Y = sin(x + π/2) = −cos(x)

Y = sin(x − π/2) = cos(x)

Step 1: Left shift π/2 → y = sin(x + π/2).

Step 2: Identity: sin(x + π/2) = cos(x).

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

Explanation

Step 1: Left shift π/2 → y = sin(x + π/2).

Step 2: Identity: sin(x + π/2) = cos(x).

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

4) A graph of y = cos(x) is shifted right by π/4. Which is correct?

Y = cos(x + π/4)

Y = cos(x − π/4)

Y = cos(π/4 − x)

Y = cos(−x − π/4)

Step 1: Right shift by π/4 means replace x with (x − π/4).

So, the final answer is y = cos(x − π/4).

Explanation

Step 1: Right shift by π/4 means replace x with (x − π/4).

So, the final answer is y = cos(x − π/4).

5) Which equation represents y = −sin(x) shifted left by π?

Y = −sin(x − π)

Y = sin(x + π)

Y = sin(x − π)

Y = −sin(x + π)

Step 1: Left shift π → y = −sin(x + π).

Step 2: Keep negative reflection.

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

Explanation

Step 1: Left shift π → y = −sin(x + π).

Step 2: Keep negative reflection.

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

6) The function y = sin(x − π/6) can be described as:

Y = sin(x) shifted left π/6

Y = sin(x) shifted right π/6

Y = sin(x) shifted up π/6

Y = sin(x) shifted down π/6

Step 1: (x − a) → right shift by a.

So, the final answer is right π/6.

Explanation

Step 1: (x − a) → right shift by a.

So, the final answer is right π/6.

7) Identify the phase shift for y = cos(2x − π).

Right π

Left π/2

Right π/2

Left π

Step 1: 2x − π = 2(x − π/2).

Step 2: Therefore, shift right by π/2.

So, the final answer is right π/2.

Explanation

Step 1: 2x − π = 2(x − π/2).

Step 2: Therefore, shift right by π/2.

So, the final answer is right π/2.

8) Which equation has the same graph as y = sin(x − π/2)?

Y = −cos(x)

Y = cos(x)

Y = −sin(x + π/2)

Y = sin(x + π/2)

Step 1: Identity: sin(x − π/2) = −cos(x).

So, the final answer is y = −cos(x).

Explanation

Step 1: Identity: sin(x − π/2) = −cos(x).

So, the final answer is y = −cos(x).

9) A sine wave has been shifted so that its maximum occurs at x = π/3, with no vertical shift and period 2π. Which equation fits?

Y = sin(x − π/3)

Y = sin(x + π/3)

Y = cos(x − π/3)

Y = cos(x − π)

Step 1: A sine’s max normally occurs at π/2 → cosine starts at max.

Step 2: Shift right π/3 → use y = cos(x − π/3).

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

Explanation

Step 1: A sine’s max normally occurs at π/2 → cosine starts at max.

Step 2: Shift right π/3 → use y = cos(x − π/3).

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

10) Consider y = sin(3x + π/3). What is the phase shift?

Left π/3

Right π/3

Left π/9

Right π/9

Step 1: Phase shift = left by φ/b = (π/3)/3 = π/9.

So, the final answer is left π/9.

Explanation

Step 1: Phase shift = left by φ/b = (π/3)/3 = π/9.

So, the final answer is left π/9.

11) Which equation represents y = cos(x) shifted left by π and reflected across the x-axis?

Y = −cos(x − π)

Y = cos(x + π)

Y = −cos(x) + π

Y = −cos(x + π)

Step 1: Left π ⇒ (x + π); reflection ⇒ negative sign.

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

Explanation

Step 1: Left π ⇒ (x + π); reflection ⇒ negative sign.

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

12) A function y = sin(x) is shifted right by π/2 and then reflected across the x-axis. Which is correct?

Y = −sin(x − π/2)

Y = −sin(x + π/2)

Y = sin(−x − π/2)

Y = −sin(−x + π/2)

Step 1: Right shift by π/2 → (x − π/2).

Step 2: Reflection → negative sign outside.

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

Explanation

Step 1: Right shift by π/2 → (x − π/2).

Step 2: Reflection → negative sign outside.

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

13) A cosine curve has its first maximum at x = 0 and the next at x = 2π, and is shifted right by π/6. Which equation matches?

Y = cos(x + π/6)

Y = cos(x − π/6)

Y = sin(x + π/3)

Y = sin(x − π/3)

Step 1: Right shift π/6 → (x − π/6).

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

Explanation

Step 1: Right shift π/6 → (x − π/6).

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

14) Which of the following is equivalent to y = cos(x + π)?

Y = −cos(x)

Y = cos(x)

Y = −sin(x)

Y = sin(x)

Step 1: Identity: cos(x + π) = −cos(x).

So, the final answer is y = −cos(x).

Explanation

Step 1: Identity: cos(x + π) = −cos(x).

So, the final answer is y = −cos(x).

15) A sine function has been horizontally shifted so that it crosses the midline going upward at x = π/4 instead of x = 0. Which equation matches?

Y = sin(x − π/4)

Y = sin(x + π/4)

Y = −sin(x − π/4)

Y = cos(x − π/4)

Step 1: Midline crossing right π/4 ⇒ right shift by π/4.

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

Explanation

Step 1: Midline crossing right π/4 ⇒ right shift by π/4.

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

16) For y = sin(2x − π/2), what is the horizontal shift?

Right π/2

Left π/2

Right π/4

Left π/4

Step 1: 2x − π/2 = 2(x − π/4).

Step 2: Shift right π/4.

So, the final answer is right π/4.

Explanation

Step 1: 2x − π/2 = 2(x − π/4).

Step 2: Shift right π/4.

So, the final answer is right π/4.

17) Which function shifts y = cos(x) right by π/3 and changes its amplitude to 2 without vertical shift?

Y = 2cos(x + π/3)

Y = 2cos(x − π/3)

Y = cos(2x − π/3)

Y = 2cos(2x − π/3)

Step 1: Right shift → (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 → (x − π/3).

Step 2: Amplitude 2 → multiply by 2.

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

18) A graph of y = sin(x) is shifted left by π/2 and vertically stretched by factor 3. Which is correct?

Y = 3sin(x − π/2)

Y = 3sin(x + π/2)

Y = sin(3x + π/2)

Y = 3sin(3x + π/2)

Step 1: Left π/2 → (x + π/2).

Step 2: Stretch by 3 → multiply by 3.

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

Explanation

Step 1: Left π/2 → (x + π/2).

Step 2: Stretch by 3 → multiply by 3.

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

19) Which equation represents a phase shift right by π/8 for y = cos(4x)?

Y = cos(4x + π/8)

Y = cos(4x − π/2)

Y = cos(4x + π/2)

Y = cos(4x − π/8)

Step 1: Right shift a ⇒ (x − a).

So, the final answer is y = cos(4x − π/8).

Explanation

Step 1: Right shift a ⇒ (x − a).

So, the final answer is y = cos(4x − π/8).

20) The function y = −sin(x + π) can be rewritten without a phase shift as:

Y = sin(x)

Y = −sin(x)

Y = sin(x + π)

Y = sin(x − π)

Step 1: sin(x + π) = −sin(x).

Step 2: −sin(x + π) = −(−sin(x)) = sin(x).

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

Explanation

Step 1: sin(x + π) = −sin(x).

Step 2: −sin(x + π) = −(−sin(x)) = sin(x).

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