How to Graph this

C

Calc12

New member
Joined
Nov 17, 2010
Messages
39
Graph a quartic polynomial function that has 4 zeros, one absolute minimum, a different local minimum and one local maximum.
 


I'm guessing that the given phrase "4 zeros" is supposed to be "4 Real zeros". Each Real zero shows up as an x-intercept; therefore, the graph must cross (or touch) the x-axis at four different places.

In pre-calculus, we learn that polynomial functions of even degree show similar global behavior. That is, the value heads in the same direction (toward either positive infinity or negative infinity), as x gets very big or very small.

This means that the ends of a fourth-degree polynomial graph must both go up or both go down. However, the function in this exercise has an absolute (global) minimum; therefore, the ends of this graph must both go up.

Many different graphs satisfy the given information. That is, the general shape is the same, but the details are different. You get to choose where the x-intercepts are located. You get to choose where to draw the minimums and maximum.

For example, you could draw the graph coming down in Quadrant II, crossing the negative x-axis, continuing down in Quadrant III, turning around and heading back over the x-axis into Quadrant II again.

At this stage, the graph shows two Real zeros and one minimum.

If the graph now turns and heads down over the x-axis and returns up a second time, there will be four x-intercepts, one maximum, and two minimums. Finish by drawing the right end of the graph heading up in Quadrant I.

Make sure that the two minimums are not at the same level. One of them must be lower (the absolute minimum).

If you need more help, please ask specific questions. If I wrote anything that you don't understand, then that's what you should ask about next.

 
oh thank you very much for your great explaination!

I was just having troubles writing out an equation that satisfies all those conditions. But I shall try again
 
Calc12 said:
Graph a quartic polynomial function that has 4 zeros, one absolute minimum,
a different local minimum and one local maximum.
Click to expand...

Suppose any of these graphs satisfying the conditions must open up.

Calc12, this quartic polynomial function could have one of the following:


\(\displaystyle 1) \ \ 4 \ imaginary \ zeroes . . . \ No \ portion \ of \ the \ graph \ ever \ crosses \ the \ x-axis.\)


\(\displaystyle 2) \ \ a \ repeated \ \ real \ zero \ . . . \ (It \ touches \ the \ x-axis\ there.) \ \ And \ it \ has \ \ 2 \ imaginary \ zeroes. \ . . .\)

\(\displaystyle (There \ is \ no \ touching/crossing \ of \ the \ x-axis \ in \ this \ portion \ of \ the \ graph.)\)


\(\displaystyle 3) \ \ 2 \ distinct \ real \ zeroes \ . . . \ (The \ graph \ crosses \ at \ 2 \ distinct \ places \ on \ the \ x-axis.) \ \\)

\(\displaystyle And \ it \ has \ 2 \ imaginary \ zeroes. . . . \ \ (In \ this \ portion, the \ graph \ does \ not \ intersect \ the\)

\(\displaystyle x-axis \ at \ all.)\)



\(\displaystyle 4) \ \ 4 \ distinct \ real \ zeroes . . . \ (The \ graph \ intersects \ the \ x-axis \ at \ 4 \ distinct \ points.)\)
 


Ah, yes; thanks lookagain. I mispoke, when I wrote that there must be four different intercepts.

There could be, but four Real zeros do not always show up as four different x-intercepts. The graph of a quartic polynomial function could touch the x-axis in two places, and not cross it anywhere. That would indicate four Real function zeros -- two zeros each with multiplicity two (i.e., repeated).

In this exercise, however, there can be at most one pair of repeated roots because both function minimums cannot be equal.

If this post is too confusing, let me know, and I'll illustrate my points with uploaded graphs.

 
Calc12 said:
I was just having troubles writing out an equation that satisfies all those conditions.

Your original post says nothing about these troubles.
Click to expand...

A fourth-degree polynomial function can take this form:

f(x) = A(x - r1)(x - r2)(x - r3)(x - r4)

where A is the leading coefficient and r1, r2, r3, and r4 are the roots of the polynomial.

As discussed above, for this exercise, the roots are all Real numbers, and they can be four distinct numbers or they can be three distinct numbers with one of them repeated.

Do you have graphing technology available?

Try different values for A, r1, r2, r3, and r4. I mean, experiment, and see what happens with different values.

Here's something else to consider. What does the sign of the leading coefficient need to be, if f(x) needs to go toward positive infinity when x is very big (positively or negatively)?

 
Calc12 said:
Graph a quartic polynomial function that has 4 zeros, one absolute minimum, a different local minimum and one local maximum.
Click to expand...

Does a different local minimum mean - only one local minimum?

Or two local minimums - out of which one is absolute minimum?
 


I'm confident that they want two different minimums, one smaller than the other.

Here's a thought, for another exercise. If both minimums were equal, it would be awkward to say, "two absolute minimums".

What could we say?

"One absolute minimum occuring twice in the domain"

"One repeated absolute minimum"

"One absolute minimum of multiplicity 2" (heh, heh)

 
You must log in or register to reply here.
Top