Questions: Write a function of the form f(x)=A sec B x for the given graph. The graph has the shape of the secant stretched vertically by a factor of 3 Thus, A=3. The graph begins at x=0 and ends at x=π/2. Thus, the period is -0 = .

Solution Steps

To determine the function \( f(x) = A \sec(Bx) \) for the given graph, we need to identify the values of \( A \) and \( B \). From the problem, we know that the graph is vertically stretched by a factor of 3, so \( A = 3 \). Next, we need to determine \( B \) by using the period of the secant function. The period of the secant function is given by \( \frac{2\pi}{B} \). Given that the graph starts at \( x = 0 \) and ends at \( x = \frac{\pi}{2} \), we can use this information to find \( B \).

1. Identify the vertical stretch factor \( A \).
2. Determine the period of the secant function from the given graph.
3. Use the period to solve for \( B \).

The graph of the function is vertically stretched by a factor of 3. Therefore, we have: \[ A = 3 \]

The graph starts at \( x = 0 \) and ends at \( x = \frac{\pi}{2} \). Thus, the period of the function is: \[ \text{Period} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

The period of the secant function \( \sec(Bx) \) is given by: \[ \text{Period} = \frac{2\pi}{B} \] Setting this equal to the calculated period: \[ \frac{2\pi}{B} = \frac{\pi}{2} \] To solve for \( B \), we can cross-multiply: \[ 2\pi = \frac{\pi}{2} B \] Multiplying both sides by 2 gives: \[ 4\pi = \pi B \] Dividing both sides by \( \pi \) (assuming \( \pi \neq 0 \)): \[ B = 4 \]

Now that we have both \( A \) and \( B \), we can write the function: \[ f(x) = 3 \sec(4x) \]

Final Answer