You are attending a baseball game. The batter hits the ball high into the air of the infield. You notice the ball, as it is hit, forms a parabolic path from contact with the bat to where it lands in the field. The path the ball traveled can be expressed by the quadratic equation y=−1/24x^2+2x , where x represents the horizontal distance and y represents the height of the ball. Interpret the following key feature of the parabola in terms of the context.

Question

GPT-4o mini · Answer

The equation of the parabola given is $ y = -\frac{1}{24}x^2 + 2x $. To interpret key features of this parabola in the context of the baseball game, we will analyze the vertex and the intercepts. ### Vertex The vertex of a parabola expressed in the standard form $ y = ax^2 + bx + c $ can be found using the formula: $$ x = -\frac{b}{2a} $$ In this case: - $ a = -\frac{1}{24} $ - $ b = 2 $ Calculating the x-coordinate of the vertex: $$ x = -\frac{2}{2 \times -\frac{1}{24}} = -\frac{2}{-\frac{1}{12}} = 24 $$ To find the corresponding y-coordinate, substitute $ x = 24 $ back into the equation: $$ y = -\frac{1}{24}(24^2) + 2(24) $$ $$ y = -\frac{1}{24}(576) + 48 $$ $$ y = -24 + 48 = 24 $$ Thus, the vertex of the parabola is at the point $ (24, 24) $. ### Interpretation The vertex $ (24, 24) $ represents the highest point of the ball's path. In the context of the baseball game, this means that the ball reaches its maximum height of 24 units (likely feet) at a horizontal distance of 24 units from the point where it was hit. This signifies the peak of the ball's trajectory during its flight, right before it begins to descend back towards the field. ### Intercepts Next, we can also find the x-intercepts (where the ball hits the ground) by setting $ y = 0 $: $$ 0 = -\frac{1}{24}x^2 + 2x $$ Factoring out $ x $: $$ 0 = x(-\frac{1}{24}x + 2) $$ This gives us one intercept at $ x = 0 $ (where the ball was hit) and the other we can solve for: $$ -\frac{1}{24}x + 2 = 0 $$ Solving for $ x $: $$ -\frac{1}{24}x = -2 \quad \Rightarrow \quad x = 48 $$ Thus, the x-intercepts occur at $ x = 0 $ (where the ball was hit) and $ x = 48 $ (where it lands). ### Summary - **Vertex:** The maximum height of the ball is 24 units at a horizontal distance of 24 units from home plate. This indicates the peak of the ball's trajectory. - **Intercepts:** The ball starts its path at $ x = 0 $ when it was hit and lands at $ x = 48 $ units, showing how far the ball traveled before touching down. These features help us understand the trajectory of the ball, including its highest point and the distance it traveled.