To create a sine function with the specified properties:

1. Minimum at (34°, -2): The phase shift and amplitude must align to achieve a minimum at this angle.
2. Equation of Axis (EOA) at 0: This means the sine wave oscillates around the horizontal line $y = 0$, so no vertical shift is needed.

The general form of a sine function is:
$y = A \sin(B(x - C)) + D$
where:

• $A$: Amplitude (distance from EOA to a peak or trough)
• $B$: Affects the period ($\text{Period} = \frac{360°}{B}$)
• $C$: Horizontal shift (phase shift)
• $D$: Vertical shift (EOA)

Key considerations:

1. The amplitude $A$ is 2 because the function oscillates between $-2$ (minimum) and $2$ (maximum).
2. The function reaches a minimum at $34°$, so the phase shift $C$ ensures this occurs.
3. Since $D = 0$, no vertical shift is applied.
4. The period $P$ can remain standard for sine, $P = 360°$, so $B = 1$.

Final Function:

$y = -2 \sin(x - 34°)$ Here, the negative sign reflects the sine wave vertically to ensure a minimum at $34°$.

Verification:

• At $x = 34°$:
  $y = -2 \sin(34° - 34°) = -2 \sin(0) = -2$
  Thus, $y = -2$, which is correct.

• The function oscillates symmetrically about $y = 0$.

Would you like a graph of this function or further clarification?

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Related Questions:

1. How do you find the phase shift of a sine function?
2. What does the amplitude of a sine function represent?
3. How is the period of a sine function calculated?
4. What modifications can be made to the equation to change the EOA?
5. How do sine and cosine functions relate in terms of transformations?

Tip:

When constructing trigonometric functions, always check key points (like maxima, minima, and zero crossings) to verify the equation.