1. The problem is to write a trigonometric function that matches the given graph. 2. The general form of a sine function is $$f(x) = A \sin(B(x - C)) + D$$ where: - $A$ is the amplitude (height of the wave), - $B$ affects the period (length of one cycle), - $C$ is the horizontal shift (phase shift), - $D$ is the vertical shift. 3. From the graph description: - The amplitude is 5, but the function is reflected vertically, so $A = -5$. - The horizontal shift is left by $\frac{\pi}{4}$, so $C = -\frac{\pi}{4}$. - There is no vertical shift, so $D = 0$. - The period is the standard sine period $2\pi$, so $B = 1$. 4. Substitute these values into the formula: $$f(x) = -5 \sin\left(1 \cdot \left(x - \left(-\frac{\pi}{4}\right)\right)\right) + 0 = -5 \sin\left(x + \frac{\pi}{4}\right)$$ 5. This matches the given function: $$f(x) = -5 \sin\left(x + \frac{\pi}{4}\right)$$ Final answer: $$f(x) = -5 \sin\left(x + \frac{\pi}{4}\right)$$