f(x) = 0.18x3 - 3.42x2 + 16.33x - 9.42

Question

Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. (Round your answers to three decimal places.)
x =       (smallest value)
x =
x =        (largest value)

Answer 1

A zero means that if we plug in a value of x into f(x) and evaluate it, f(x) should be zero.  Since we need to find approximate zeros.  We need to find x values so when f(x) is evaluated, f(x) is close to zero.  Now, if we use the rational root theorem to guess the zeros, the possible zeros will be ±0.1, ± 0.3, and ±0.9.

 

 

We will evaluate f(x) at all of these possible roots.

 

f(-0.9) = -27.018

 

f(-0.3) = -14.632

 

f(-0.1) = -11.087

 

f(0.1) = - 7.821

 

f(0.3) = -4.823

 

f(0.9) = 2.638

 

 

We have tried all the roots.  Notice that as x increases, f(x) increases from a negative value.  Between x=0.3 and x=0.9, there is a zero because f(x) changes signs.

 

 

Let's evaluate f(0.5) to f(0.8) to see if we get a value close to zero.

 

f(0.5) = -2.088

 

f(0.6) = -0.814

 

f(0.7) = 0.397

 

f(0.8) = 1.547

 

 

So between x=0.6  and x=0.7, there is a zero.  The zero is more likely close to x=0.7.   We can approximate that one zero is within this interval.  There should be two more zeros because the degree of f(x) is 3.  This means we expect to get 3 zeros.

 

 

Based on the shape of a degree 3 polynomial, as f(x) starts to increase, it will cross the y-axis at (0, -9.42), then at some point, decrease, and then increase.  This means that the zeros start between x=0.6 and x=0.7.  We will find more zeros as we read the graph towards the right.  Therefore, you would use an x interval of

 

x=0  to x=20    to find the zeros.  Use the graphing calculator to find the other zeros.  

 

To change the window, go to WINDOW, and set these parameters:

 

Xmin = 0

Xmax = 20

 

 

Next, go to Y= and type in the function you are given.  Then press GRAPH.  You will see you graph being constructed.  Then press 2ND, TRACE, and chose the zero option.  It will allow you to find all the zeros.  You will need to guess the x values in order for the calculator to give you results.

 

 

Follow these steps and you will have your solution.

Answer 2

Blake,

I assume there is a table that you were using to approximate the zeros of the function...? I used my graphing calculator utility to find the max, f(3.192) = 13.714 (rounded to three decimal places) and the min, f(9.475) = -8.616 (rounded to three decimal places). 

 

I hope that helps!

 

Neal