Write the quadratic function \( f(x)=-6(x-1)^{2}+24 \)

To rewrite the given quadratic function, let's expand it. The function \( f(x) = -6(x - 1)^{2} + 24 \) can be expanded by first applying the square: 1. \( (x - 1)^{2} = x^{2} - 2x + 1 \) 2. Now substituting back in gives us: \( f(x) = -6(x^{2} - 2x + 1) + 24 \) 3. Distributing the -6: \( f(x) = -6x^{2} + 12x - 6 + 24 \) 4. Combining like terms results in: \( f(x) = -6x^{2} + 12x + 18 \) So, the quadratic function in standard form is: \( f(x) = -6x^{2} + 12x + 18 \). Now, how about some interesting facts? First, did you know that quadratic functions like this one can form beautiful parabolas when graphed? The vertex of the parabola, which for this function is at the point (1, 24), is actually the highest point on the graph since the coefficient of \( x^2 \) is negative, indicating it's a "downward-opening" parabola. Second, quadratic functions frequently model real-world scenarios such as projectile motion. For example, if you throw a ball in the air, the height of the ball over time can be represented by a quadratic equation, helping you determine its peak height and how long it stays in the air before falling back down!