Inferring properties of polynomial function from its graph.

Question

 

 Below is a graph of a polynomial function ƒ with real coefficients. Use graph to answer following questions about ƒ.  All local extreme of ƒ are shown on graph.

 

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a. the function is decreasing over which intervals?  Choose all that apply.   1. (-∞,-5) 2. (-5,-3)

3. (-5,0) 4. (0,3) 5. (6,8) 6.(8,∞)

b. The function f has local maxima at which x-values?  If there is more than one value, separate with comas.

c. What is the sign of the leading coefficient of?  1. positive 2. negative or 3. not enough info

d. Which of the following is a possibility for the degree of f?  Choose all that apply.  a.4 b.5  c.6 d.7 e.8 f.9


I have already answered questions      a. b, & c correctly.  Question d was answered incorrectly (5,7,9,)

 

I have the graph but don't know how to add it on this form.

Thanks for any help

 

Answer 1

You don't have a graph here, but I can still guide you on how to tackle this problem.  Since you have no problem with a, b, and c, lets just jump to d.

 

 

The shape of the graph can help in determining the degree of f.  The degree is usually the highest exponent of x in the function.

 

If the graph has a symmetrical shape, then the graph has an even degree.

If the graph has an asymmetrical shape, then the graph has an odd degree.

 

Sometimes, the number of extremas and number of times the graph crosses the x-axis can help in determining the degree of the function, but not always the case.

 

 

In theory:

 

If the function has n local extremas, then the degree of the function is (n + 1).

If the function crosses the x-axis t exactly n times, then the degree of the function is n.

 

 

However, there are cases in which a degree 4 function will cross the x-axis at most 2 times, and have at most 1 local extrema.

 

 

This is due to how the function is written.

 

If your function for example is   f(x) = x⁴ + 3x³ + 4x² - x - 8 , then the theory holds.

If your function for examples is  f(x) = x⁴ + 3x² - 8 , then the theory is not so true.  Notice that this function is missing a  x³ and x term.

 

 

So when you examine your graph, you need to look at the shape first.  Then apply the other theories to make an educated guess to determining the degree of your function.