Question

Let \(f(x) = \ln(5x) - \sin\left(\frac{\pi x}{2}\)). Find the vertex form of the function.

Answer 1

The vertex form of the function f(x) = ln(5x) - sin(πx/2) is f(x) = ln(5) + √2 sin(πx/2 - π/4).

Step-by-step explanation:

To find the vertex form of the function f(x) = ln(5x) - sin(πx/2), we focus on rewriting the trigonometric part in a more manageable form while the logarithmic term, ln(5x), remains unchanged.

Starting with the trigonometric term -sin(πx/2), we factor out the coefficient -1 for simplification purposes:

-sin(πx/2) can be expressed as -√2 * (√2/2) * sin(πx/2).

Completing the square within the sine term involves adding and subtracting π/4:

= -√2 sin(πx/2 - π/4).

Thus, the vertex form of the function becomes f(x) = ln(5) + √2 sin(πx/2 - π/4). This form facilitates the identification of the amplitude, period, and phase shift of the trigonometric component, providing valuable insights into the function's behavior and characteristics.