11. Vertex: \( (5,-3) \); passes through the point \( (1,5) \)

To determine the equation of the parabola that has a vertex at \( (5, -3) \) and passes through the point \( (1, 5) \), you can start with the vertex form of a parabola's equation: \[ y = a(x - h)^2 + k \] Here, \( h = 5 \) and \( k = -3 \), so the equation becomes: \[ y = a(x - 5)^2 - 3 \] Now plug in the point \( (1, 5) \) to solve for \( a \): \[ 5 = a(1 - 5)^2 - 3 \] \[ 5 = a(16) - 3 \] \[ 8 = 16a \] \[ a = \frac{1}{2} \] So, the equation of the parabola is: \[ y = \frac{1}{2}(x - 5)^2 - 3 \] This tells us that the parabola opens upwards and reaches its minimum at the vertex \( (5, -3) \). The beauty of parabolas is that they're not just numbers on a piece of paper! They are all around us—in the design of satellite dishes, the trajectory of basketball hoops, and even in the shape of some fun roller coasters. Understanding their mathematical properties can unveil the secrets of these exciting real-world applications. Moreover, it's essential to remember that when solving for parameters like the vertex or equations from points, minor arithmetic errors can lead to significant changes in the final result. Double-checking calculations and ensuring accurate substitution can help stave off common mistakes, ensuring that those parabolas reflect the true beauty of their curves!