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Chapter 3: Problem 83

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-4 x^{2}+20 x+160$$

Short Answer

Expert verified
The vertex for the parabola y=-4 x^{2}+20 x+160 is (2.5,155). Because a is negative, the parabola opens downward.

Step by step solution

01

Identify a, b and c

First, identify the values of a, b and c in the given quadratic equation, which are -4, 20 and 160 respectively.
02

Find the x-coordinate of the vertex

The x-coordinate of the vertex of a parabola given by an equation y = ax^{2} + bx + c is given by -b/(2a). So here, x = -b/(2a) = -20/(2*-4) = 2.5
03

Find the y-coordinate of the vertex

The y-coordinate of the vertex can be found by substituting the x-coordinate into the equation. So, y = -4*(2.5^2) + 20*2.5 + 160 = 155.
04

Determining the direction in which the parabola opens

The parabola opens downward because \( a \), the coefficient of \( x^{2} \) is negative (-4).
05

Graphing the function

Plot the vertex point (2.5, 155) on the graph. Considering that the parabola opens downward, sketch the parabola based on this information.

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