Expert: Steve Holleran - 3/14/2008

Question
QUESTION: hi

how can i find the graph & the equation for these two statements:

- An odd-degree function with 3 relative maximum, 3 relative minimum and negative end behavior on the right.


- An even-degree function with 4 relative maximum, 3 relative minimum.


ANSWER: Hi Meme,

For the first one, let's say you give it zeros at

x= -5, -3, -1, 1, 3, 5, 7.

Then an equation could be

f(x) = - (x+5)(x+3)(x+1)(x-1)(x-3)(x-5)(x-7)

This would make it odd degree, and the - in front would give it "down right" end behavior.  There would be local max points between
-3 and -1, 1 and 3, 5 and 7 ;

and local min points between
-5 and -3, -1 and 1, 3 and 5


For the second one, give it zeros at x = -5, -3, -1, 1, 3, 5, 7, 9

Then an equation could be

f(x) = -(x+5)(x+3)(x+1)(x-1)(x-3)(x-5)(x-7)(x-9)

This would be 8th degree and have down left, down right end behavior,
with
local max points between -5 and -3, -1 and 1, 3 and 5, 7 and 9

and local min points between  -3 and -1, 1 and 3, 5 and 7

I hope this is what you needed

Steve


---------- FOLLOW-UP ----------

QUESTION: thank you very much for your answer.

when i tried to plot the graph, i couldn't bacuse the numbers are very big. Can you please give me smaller numbers so that i can plot the graph.

Answer
Hi,

Well the only way you're going to get smaller values is to take very small x-values.  I didn't know you needed to plot the exact points.  An eighth-degree polynomial is just not going to give nice, small values, especially under the conditions you describe.

You could try taking x-values like .1, .4. , .6 etc, but the numbers still would not come out "nice".

Steve