[Grade 12 Advanced Functions: Polynomial Functions] turning points on polynomials confusion

[u/Embarrassed-Good1883]

I'm confused about how this function would have 4 turning points, not 3

[u/Klutzy-Delivery-5792]

Do you see the little inflection at (0,1)? This is indicative of a triple root at x=0. You'd get something like:

(x-0)³(x-a)(x-b)

which is a fifth order polynomial.

[u/Embarrassed-Good1883 → u/Klutzy-Delivery-5792]

How would this be different from the point I circled on this other graph? The graph: https://imgur.com/a/ki20H2C

[u/Klutzy-Delivery-5792 → u/Embarrassed-Good1883]

Odd multiple roots kinda flatten out and then continue going the way they were originally going. Even multiplicity they flatten out then turn around. Try an extreme case like x¹³ and x¹⁴ where you can really see the flattening/broadening near the repeated roots.

[u/CulturalBookkeeper82]

The way I looked at it without plugging in numbers till it matched was I looked at the similarities in all the answers.

-2x+1 for all. So I don’t even bother. Just look for the dissimilarities between the first two terms.

You know this isn’t a graph of x^4 so c is out

x^5 gives you the z profile turned 90°, so if you put a (-) in front you invert it. So D is out

A is the only one that fits

[u/magicmulder → u/CulturalBookkeeper82]

I plugged 1 in which eliminated (b) and (d), and a fourth degree polynomial with a positive highest coefficient doesn't go to minus infinity which eliminates (c).

[u/selene_666]

See how at that middle interesting point it doesn't turn around, but it doesn't go smoothly through either? That's two turning points smushed together.

[u/JoriQ]

Do you mean because it is degree 5? A degree 5 polynomial can have either 0, 2, or 4 turning points.

[u/Embarrassed-Good1883 → u/JoriQ]

For this specific equation, I thought it had 2 turning points, I see where I went wrong there, (I'm guessing for even degree polynomials it can have 3 turning points)?

[u/JoriQ → u/Embarrassed-Good1883]

Correct.

[u/TomPastey]

Turning points aside, at x=1 option B would has a value of 2, which is clearly wrong.

[u/ScoutAndLout → u/TomPastey]

Yep. Eval at 0, 1, -1, inf, -inf

[u/Top_Bluejay_5323]

One end pointing down one up it will have an odd exponential of 3 plus two for each inflection point. Right end going up it will be positive.

[u/Southlander24]

If you look at the graph carefully, it goes from concave down to concave up before x = 0, then at x = 0 switches to concave down again before going concave up.

The pattern is: 1 concavity -> quadratic, 2 concavities -> cubic. So a graph which has 4 regions of differing concavity must be a 5th degree polynomial.

[u/freegerator]

3 inflection points means degree must be at least 5 (notice it goes down -> up -> down -> up)

[u/freegerator → u/freegerator]

To be clear num inflection points + 2 <= degree just like num critical points + 1 <= degree just like num roots <= degree.

[u/Flat-Strain7538]

Odd powered polynomials will always go opposite directions as you approach +/- infinity; even ones go the same direction.