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Chapter 2: Problem 85

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

Short Answer

Expert verified
The vertex of the parabola is at (80, 800). After plotting this on a graphing utility with a viewing rectangle of x = [50, 110] and y = [0, 1000], it shows a downward opening parabola.

Step by step solution

01

Find the vertex

We find the x-coordinate of the vertex using the formula \(h=-\frac{b}{2a}\). Here, in the function \(y = -0.25x^2 + 40x\), the values of \(a\) and \(b\) are -0.25 and 40, respectively. So: \[h=-\frac{40}{2*-0.25} = 80\] The y-coordinate is found by plugging the x-coordinate into the function: \[k = -0.25*80^2 + 40*80 = 800\] Therefore, the vertex is at (80, 800)
02

Determine a reasonable viewing rectangle

A reasonable viewing rectangle can be defined by taking a range of values for x that includes the x-coordinate of the vertex and extending a bit to each side, and a range of y-values that includes the y-coordinate of the vertex and also extends above and below. In this case, a reasonable viewing rectangle might be x = [50, 110] and y = [0, 1000].
03

Plot the graph

With the viewing rectangle and the vertex, we are ready to plot the graph of the function. The graph should be a downward opening parabola with the vertex at (80, 800). On a graphing utility, the quadratic function will be plotted using the determined interval for x and y values. The graph will confirm the nature and direction of the parabola.

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Most popular questions from this chapter

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=3,\) a horizontal asymptote \(y=0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

The perimeter of a rectangle is 50 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 114 square feet.

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-1}{x}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I began the solution of the rational inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero.

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?
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