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Chapter 2: Problem 82

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. $$f(x)=-4 x^{3}+4 x^{2}+15 x$$

Short Answer

Expert verified
The graph of \(f(x)=-4 x^{3}+4 x^{2}+15 x\) begins from the top, passes through the x-axis at the real zeros of the function and certain other solution points, and finally points downwards as predicted by the Leading Coefficient Test

Step by step solution

01

Leading Coefficient Test

The leading coefficient test helps in predicting the behavior of the polynomial when \(x\) approaches positive or negative infinity. The leading term of the given function is \(-4x^{3}\). Since the degree \(3\) is odd and the leading coefficient \(-4\) is negative, the right end of the graph will point downwards while the left end will point upwards.
02

Finding the real zeros

The real zeros of the function can be found by setting \(f(x)\) equal to zero. Hence, start by setting \(-4 x^{3}+4 x^{2}+15 x = 0\). We can factor out \(x\) from each term to get the equation in the form \(x(-4x^{2}+4x+15)=0\). Therefore, the real zeros of the equation will be \(x=0\) and \(x\) equals to the roots of the quadratic equation \(-4x^{2}+4x+15=0\).
03

Plotting sufficient solution points

By choosing some additional points for \(x\) and calculating the corresponding \(f(x)\) values, we can obtain a more accurate graph. It's important to be mindful of the chosen points; mantaining variety helps to create a better representation.
04

Drawing a continuous curve

Once all the roots and some additional solution points are plotted on the graph, a continuous curve can be drawn. The curve should pass through all the plotted points and adhere to the behavior as predicted by the Leading Coefficient Test. The graph should start from the top (as \(x\) approaches negative infinity), pass through all the points and end towards the bottom (as \(x\) approaches positive infinity).

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Most popular questions from this chapter

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=3 x^{3}+2 x^{2}+x+3$$

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