If the height that a toy airplane can fly is given by the following expression: \( -3 x^{2}-5 x+9 \). Determine the maximum height that the airplane can fly.
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To find the maximum height of the toy airplane represented by the quadratic equation \( -3x^{2}-5x+9 \), we can use the vertex formula, which gives the x-coordinate of the vertex as \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = -5 \). Calculating gives us: \[ x = -\frac{-5}{2 \times -3} = \frac{5}{-6} = -\frac{5}{6}. \] Next, we substitute this value back into the original expression to find the maximum height: \[ -\left(3\left(-\frac{5}{6}\right)^{2} + 5\left(-\frac{5}{6}\right) + 9\right). \] Calculating this results in a maximum height of \( 10.25 \). This means the toy airplane can soar to a fantastic height of \( 10.25 \) units! It's important to note that maximizing the height of a quadratic function involves understanding its parabola shape, which opens downwards when the leading coefficient (a) is negative. This dynamic behavior of quadratics can be key in various real-world scenarios, such as predicting projectile motion or optimizing design features in engineering.
