Final answer:
The maximum height of the ball is 9 feet, achieved at 6 feet horizontally from the starting point. The total horizontal distance the ball travels is 12 feet, as determined by the quadratic function modeling its parabolic path.
Step-by-step explanation:
The student wants to find the maximum height of a ball kicked into the air and the total horizontal distance it travels, following the function y = −0.25x² + 3x, where x is the horizontal distance in feet and y is the vertical distance in feet.
To find the maximum height, we need to determine the vertex of the parabola, as the vertex represents the highest point on the graph of a downward-opening parabola. In the given quadratic equation, the coefficient of x² is negative, indicating that the parabola opens downwards. The x-coordinate of the vertex of a parabola in the form y = ax² + bx + c is found using -b/(2a). Here we have a = -0.25 and b = 3, so the x-coordinate is -3/(2 × -0.25) = 6 feet. Substituting x = 6 into the equation yields y = −0.25(6)² + 3(6) = 9 feet, which is the maximum height of the ball.
To find the total horizontal distance traveled, we need to find the roots of the equation, also known as the x-intercepts, where the ball hits the ground (y = 0). This is equivalent to solving 0 = −0.25x² + 3x. Factoring out an x gives x (−0.25x + 3) = 0. The roots are x = 0 and x = 12 feet. Since the ball starts at x = 0, the total horizontal distance is 12 feet.