Problem 83 - 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. $$y=-4 x^{2}+20 x+160$$

Step-by-Step Solution

Answer

The vertex of this parabola is at (2.5,172.5). A reasonable viewing rectangle for this function on a graphing calculator might be: Xmin = 0, Xmax = 5, Ymin = 150, Ymax = 180.

Step 1: Identify the vertex of the parabola

The vertex of a parabola given by a quadratic equation in the standard form \(y=ax^2+bx+c\) can be found using the formula \(h=-b/(2a)\). Substituting the values for 'a' and 'b' from the given equation, we find the x-coordinate of the vertex, \(h\), to be \(h=-20/(2*-4)=2.5\). To find the y-coordinate of the vertex, substitute the x-coordinate back into the equation to find \(k=-4*(2.5)^2+20*2.5+160=172.5\). Therefore, the vertex of the parabola is at (2.5,172.5).

Step 2: Determine a reasonable viewing rectangle

A reasonable viewing rectangle on a graphing utility should include the vertex of the graph, and also give a good view of the shape of the parabola. Keeping in mind that this is a downward facing parabola (since 'a' is negative), a reasonable viewing rectangle might have these values: Xmin = 0, Xmax = 5, Ymin = 150, Ymax = 180.

Step 3: Graph the function

Using your graphing utility, input the equation of the parabola \(y=-4x^{2}+20x+160\). Use the viewing rectangle determined in the previous step to view the graph. You should see a downward-facing parabola with vertex at (2.5,172.5).