Using a certain ramp at the skate park, a skater’s height off the ground, in feet, is modeled by the equation y=0.25x^2 −4x+19

Question

Using a certain ramp at the skate park, a skater’s height off the ground, in feet, is modeled by the equation  y=0.25x^2 −4x+19
 , where x is the skater’s horizontal distance from the jumping platform. How far from the jumping platform is a skateboarder when they are at their minimum height off the ground?

GPT-4o mini · Answer

To find the horizontal distance $ x $ from the jumping platform at which the skateboarder is at their minimum height, we need to determine the vertex of the given quadratic equation: $$ y = 0.25x^2 - 4x + 19 $$ For a quadratic equation in the form $ ax^2 + bx + c $, the x-coordinate of the vertex (which gives the minimum or maximum point) can be found using the formula: $$ x = -\frac{b}{2a} $$ In our equation, $ a = 0.25 $ and $ b = -4 $. We can substitute these values into the vertex formula: $$ x = -\frac{-4}{2 \times 0.25} $$ Calculating inside the equation: $$ x = \frac{4}{0.5} = 8 $$ Thus, the skateboarder is **8 feet** from the jumping platform when they are at their minimum height off the ground.