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Chapter 2: Problem 78

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the real zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. $$g(x)=-x^{2}+10 x-16$$

Short Answer

Expert verified
The graph of the function \(g(x)=-x^{2}+10 x-16\) is a downward-opening parabola with its vertex at \(x=5\) and roots at \(x=2, x=8\).

Step by step solution

01

Apply the Leading Coefficient Test

In the Leading Coefficient Test, look at the leading term of the polynomial, which over here is \(-x^{2}\). Specifically, look at the coefficient of this leading term, which here is -1. Since the coefficient is negative, the graph opens downwards. This is crucial for sketching the graph later.
02

Find the real zeros of the polynomial

Next, proceed to find the real zeros of the polynomial \(g(x)=-x^{2}+10 x-16\). These are the x-values for which \(g(x)=0\). This leads to the equation \(-x^{2}+10 x-16=0\). This equation can be solved using factorization or the quadratic formula. In this case, rewriting the equation as \(x^{2}-10x+16=0\) and then, factorizing to obtain \((x-8)(x-2)=0\), find the roots, \(x=2, x=8\). These are the zeros or roots of the polynomial which are going to be crucial points in the graph.
03

Plot Solution Points and Draw Continuous Curve

On the coordinate plane, plot the roots or zeros obtained in the previous step, which are \(x=2\) and \(x=8\). Recall from Step 1 that the graph opens downwards, use this along with the roots to sketch the graph. The vertex (the point at which the curve changes direction) of the graph will be at the midpoint between the two roots. Since the graph opens downwards, the vertex will also be the maximum point of the parabola, which occurs at \(x=\frac{2+8}{2}=5\). You have sufficient points to sketch the parabola at this point, then trace the graph through the plotted points and extend it in the directions indicated by the Leading Coefficient Test.

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Most popular questions from this chapter

Use the given zero to find all the zeros of the function. Function \(h(x)=3 x^{3}-4 x^{2}+8 x+8\) Zero \(1-\sqrt{3} i\)

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.

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Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{2}-x+56$$
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