How do you sketch this sort of graph? (2 Viewers)

[BlueGas]

Let's say the graph has stationary points, (0,0) and (3,5). By the way, point (3,5) is a maximum, so how would you sketch a graph like this? I'm looking for answers that include pictures, thanks.

 

[Librah]

Let's say the graph has stationary points, (0,0) and (3,5). By the way, point (3,5) is a maximum, so how would you sketch a graph like this? I'm looking for answers that include pictures, thanks.

What degree is this polynomial? Also are there any other roots?

 

[BlueGas]

What degree is this polynomial? Also are there any other roots?

lol, I don't know what you're trying to say...

 

[PhysicsMaths]

Let's say the graph has stationary points, (0,0) and (3,5). By the way, point (3,5) is a maximum, so how would you sketch a graph like this? I'm looking for answers that include pictures, thanks.

The possibilities are virtually endless, but here's a typical example of what it might look like:
http://imgur.com/gvV6rez

Also note, that since (3,5) is a max sp,it is guaranteed that (0,0) is a min.
Also between those two points exist a pt of inflexion (change in concavity)

Other than that, it can be anything. Not enough information is given

 

[PhysicsMaths]

lol, I don't know what you're trying to say...

What he is asking is for the power of the leading term of the polynomial, e.g. x^2, x^3...
Roots are where the curve hits the x-axis, so for my example, double root at (0,0) and single root at approx. x = 5

Single roots just cross the x-axis
Double roots sort of bounce back up
And triple roots are like pts of inflexion

 

[BlueGas]

The possibilities are virtually endless, but here's a typical example of what it might look like:
http://imgur.com/gvV6rez

Also note, that since (3,5) is a max sp,it is guaranteed that (0,0) is a min.
Also between those two points exist a pt of inflexion (change in concavity)

Other than that, it can be anything. Not enough information is given

See that's the part that confuses me, how can (0,0) be a min? It's 0, it's neither a maximum or minimum. How come the answer for this type of question says that (0,0) is a horizontal point of inflexion?

 

[Librah]

See that's the part that confuses me, how can (0,0) be a min? It's 0, it's neither a maximum or minimum. How come the answer for this type of question says that (0,0) is a horizontal point of inflexion?

... it's a double root. the polynomial lets say p(x)= x^2 multiplied by (......whatever else cause we don't know based on your info). If your saying it's a min point, how can it be a horizontal point of inflexion? By definition that's impossible.

 

[BlueGas]

... it's a double root. the polynomial lets say p(x)= x^2 multiplied by (......whatever else cause we don't know based on your info). If your saying it's a min point, how can it be a horizontal point of inflexion? By definition that's impossible.

Here it is, it's question 5 on HSC Mathematics 2003.

(a) Consider the function f(x) = x^4 − 4x^3

(ii) Find the coordinates of the stationary points of the curve y = f(x), and
determine their nature.
(iii) Sketch the graph of the curve y = f(x), showing the stationary points.

Answer: (0,0) is a horizontal point of inflexion and (3, -27) is a minimum. How would you sketch this?

 

[Librah]

Here it is, it's question 5 on HSC Mathematics 2003.

(a) Consider the function f(x) = x^4 − 4x^3

(ii) Find the coordinates of the stationary points of the curve y = f(x), and
determine their nature.
(iii) Sketch the graph of the curve y = f(x), showing the stationary points.

Answer: (0,0) is a horizontal point of inflexion and (3, -27) is a minimum. How would you sketch this?

Well why'd u say 0,0 was a min point... I don't know how to use this latex thing, so just go wolframalpha the eq into the system to see graph, if it's still for free

 

[PhysicsMaths]

http://imgur.com/FcHMGLH
This is what it should look like

Note that since (3,-27) is a minimum stat. pt., the graph must go up after that
and btw, f(x) can be factorised to x^3(x-4)
and the x^3 represents the triple root (pt of inflexion) at (0,0)
After differentiating, (0,0) is also a stat. pt., so (0,0) is a horizontal pt of inflexion

 

[BlueGas]

Well why'd u say 0,0 was a min point... I don't know how to use this latex thing, so just go wolframalpha the eq into the system to see graph, if it's still for free

Where did I say (0,0) was a minimum?

 

[Librah]

Where did I say (0,0) was a minimum?

Hmm must have misread, my bad. But it should pass through 0,0 and have a cubic-like appearance and pass through x=4 after a min point (3,-27). Oh right this part "point (3,5) is a maximum" was this apart of the same question? Cause there are only 2 stationary points in your graph and one of them is at (3,-27)

 

[BlueGas]

Hmm must have misread, my bad. But it should pass through 0,0 and have a cubic-like appearance and pass through x=4 after a min point (3,-27)

If (3, -27) was a maximum, how would that look like? Or is that not possible?

 

[Librah]

If (3, -27) was a maximum, how would that look like? Or is that not possible?

You would have to completely change the equation. Assuming 0,0 was still your horizontal inflex point. You would essentially either have a curve with discontinuities or 2 or more stationary points. Better ask one of the maths pro's on this site though, i havn't done anything for the past 2-3 months.

 

[BlueGas]

You would have to completely change the equation. Assuming 0,0 was still your horizontal inflex point. You would essentially either have a curve with discontinuities or 2 or more stationary points.

Okay thanks for your help, even though PhysicsMaths was the one who provided the best answer including a picture.