Find the equation of the curve for the given slope and point through which it passes. Use a graphing calculator to display the curve. Slope given by \( 4 x / y \); passes through \( (5,2) \) What is the equation of the curve? \( y^{2}=\square \)

To find the equation of the curve given the slope \( \frac{4x}{y} \) and the point \( (5,2) \), we start by rewriting the slope as a differential equation. That gives us: \[ y \frac{dy}{dx} = 4x. \] Next, we separate the variables by rearranging to: \[ y \, dy = 4x \, dx. \] Now, we can integrate both sides: \[ \int y \, dy = \int 4x \, dx, \] \[ \frac{y^2}{2} = 2x^2 + C. \] Multiplying through by 2, we get: \[ y^2 = 4x^2 + 2C. \] To find \( C \), we can use the point \( (5, 2) \): \[ 2^2 = 4(5^2) + 2C \] \[ 4 = 100 + 2C \] \[ 2C = 4 - 100 = -96 \] \[ C = -48. \] Thus, substituting \( C \) back into the equation, we arrive at: \[ y^2 = 4x^2 - 96. \] So the final equation will be: \[ y^{2} = 4x^{2} - 96. \] In this case, you can graph the equation to visualize the curve! Just plug in the equation and see how it unfolds on your graphing calculator. Enjoy exploring the curve! As for shading beyond the mathematics, did you know that curves described by differential equations can represent real-world phenomena like population growth or the trajectory of projectiles? The beauty of math often lies in its connections to nature! And if you're interested in diving deeper, there are countless resources on differential equations that can shine a light on their applications. Pick up a book on elementary differential equations, and you’ll uncover a treasure trove of math magic that applies everywhere from physics to economics!