Which of the following best represents the direction field for the differential equation $$\frac{dy}{dx} = -\frac{x}{4y}$$? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2020 - Paper 1

To sketch the direction field, we need to understand that the slope of the solution curves at any point $(x, y)$ is given by the equation.

• When $y > 0$, the slopes will be negative for positive $x$, indicating falling curves.
• When $y < 0$, the slopes will be positive for positive $x$, indicating rising curves.
• The slopes approach zero as $y$ approaches zero.

Observing the provided graphs:

• Graph A shows the slopes consistent with the described behavior: negative for $y > 0$ and positive for $y < 0$ as $x$ increases.
• The other graphs do not properly reflect the characteristics outlined, as their direction fields do not match well with the expected behavior of the slopes.

Therefore, the correct option that best represents the direction field for the given differential equation is A.