Problem 82 - College Algebra

Question

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-0.25 x^{2}+40 x$$

Step-by-Step Solution

Answer

The vertex of the parabola is \(80, 800\). The reasonable viewing rectangle for graphing the quadratic function is \([0, 160] x [0, 1000]\). The graph is a downward-opening parabola with the vertex at \(80, 800\) and intercepts the x-axis at \(x = 0\) and \(x = 160\).

Step 1: Find the vertex of the parabola

First, the vertex of the parabola is calculated using the formula \(-b/2a, f(-b/2a)\). Here, \(a = -0.25\) and \(b = 40\). Substituting these values into the formula, the x-coordinate of the vertex is found to be \(-b/2a = -(40)/2(-0.25) = 80\). Substituting \(x = 80\) into the quadratic equation gives \(y = -0.25*80^2 + 40*80 = 800\). So, the vertex of the parabola is \(80, 800\).

Step 2: Determine a reasonable viewing rectangle

A reasonable viewing rectangle can be chosen by considering the vertex and the direction of the parabola. Since the coefficient of \(x^2\) is negative, the parabola opens downwards. Therefore, an appropriate viewing rectangle might be \([0, 160] x [0, 1000]\). This rectangle includes the vertex and also gives a sufficient view of the graph of the parabola.

Step 3: Graph the quadratic function

Next, graph the quadratic function using a graphing utility. The graph should be a parabola that opens downward, with the vertex at \(80, 800\) and intercepts the x-axis at \(x = 0\) and \(x = 160\).