Graphing Quadratic Functions: How to Find Critical Points and Inflection Points

[MAC5494]

Homework Statement

I've been looking at the problem for a while and I just can't figure it out. It is graph sketching.

[STRIKE]y=x²(x²-4)²[/STRIKE]
the correct equation is this:
y=x²(x-4)²

Homework Equations

Solve for critical points, and inflection points in order to get points needed to graph function.

The Attempt at a Solution

[STRIKE]y=x²(x-4)²[/STRIKE]
y=x²(x²-4)²
y=x⁴-8x³+16x²

y'=4x³-24x²+32x
0=4x(x-4)(x-2)
x=0,2,4

local min: 0,4
local max: 2

y"=12x²-48x+32

I don't know what to do to solve for the inflection points.

 

[vhbelvadi]

I don't know what to do to solve for the inflection points.

Well, inflection points can be arrived at by solving f "(x)=0 and finding values of x and substituting them again in the given equation.
That should give you all inflection points of that function.

 

[MAC5494]

Well, inflection points can be arrived at by solving f "(x)=0 and finding values of x and substituting them again in the given equation.
That should give you all inflection points of that function.

Yeah I'm aware of that, and I've tried, but I can't figure out how to solve for x.

 

[FeDeX_LaTeX]

Yeah I'm aware of that, and I've tried, but I can't figure out how to solve for x.

Are you asking us how you might solve $12x^2 - 48x + 32 = 0$?

 

[MAC5494]

Are you asking us how you might solve $12x^2 - 48x + 32 = 0$?

Yes, or at least some pointers in the right direction. I'd like to try to do it myself if possible.

 

[FeDeX_LaTeX]

Yes, or at least some pointers in the right direction. I'd like to try to do it myself if possible.

Have you seen the quadratic formula before?

In relation to sketching the original curve, it might save you some time to note whether or not the function is odd or even.

 

[MAC5494]

Have you seen the quadratic formula before?

In relation to sketching the original curve, it might save you some time to note whether or not the function is odd or even.

Yes! I didn't even think about that! Thank you for your help! It should be an even function just by looking at the formula. How do you figure out algebraically if it is even or odd?

 

[FeDeX_LaTeX]

Yes! I didn't even think about that! Thank you for your help! It should be an even function just by looking at the formula. How do you figure out algebraically if it is even or odd?

In general, a function $f(x)$ is odd if $f(-x) = -f(x)$, and even if $f(-x) = f(x)$. (You can also decompose a function into an even part and an odd part, but that's irrelevant here.)

What happens when you replace $x$ with $-x$ in your function?

 

[MAC5494]

In general, a function $f(x)$ is odd if $f(-x) = -f(x)$, and even if $f(-x) = f(x)$. (You can also decompose a function into an even part and an odd part, but that's irrelevant here.)

What happens when you replace $x$ with $-x$ in your function?

f(x) = x⁴-8x³+16x²

f(-x) = (-x)⁴ -8(-x)³+16(x)²

= x⁴ + 8x³ + 16x²

So I was wrong. It's actually neither even or odd.

 

[SteamKing]

From the OP:
y = x^{2}(x^{2}-4)^{2}
y = x^{4}-8x^{3}+16x^{2}

You need to work on your basic algebra skills: the first equation does not lead to the second.

y = x^{2}(x^{2}-4)^{2}
y = x^{2}(x^{4}-8x^{2}+16)
y = (x^{6}-8x^{4}+16x^{2})

 

[MAC5494]

From the OP:
y = x^{2}(x^{2}-4)^{2}
y = x^{4}-8x^{3}+16x^{2}

You need to work on your basic algebra skills: the first equation does not lead to the second.

y = x^{2}(x^{2}-4)^{2}
y = x^{2}(x^{4}-8x^{2}+16)
y = (x^{6}-8x^{4}+16x^{2})

Oh god. Well there's my issue. I forgot to foil. Thanks for your help. I don't think I'll have an issue with this problem anymore. That's embarrassing.

 

[Ray Vickson]

f(x) = x⁴-8x³+16x²

f(-x) = (-x)⁴ -8(-x)³+16(x)²

= x⁴ + 8x³ + 16x²

So I was wrong. It's actually neither even or odd.

The function is even; x^2 is even and x^2 - 4 is even, so x^2 * (x^2-4)^2 is even. No work needed!

 

[MAC5494]

The function is even; x^2 is even and x^2 - 4 is even, so x^2 * (x^2-4)^2 is even. No work needed!

I forgot to foil so I came out with the wrong function.

 

[FeDeX_LaTeX]

I forgot to foil so I came out with the wrong function.

Yes, but you can tell that the function is even without any expansion, since you have a product of two even functions.

 

[MAC5494]

Yes, but you can tell that the function is even without any expansion, since you have a product of two even functions.

Yeah I know. I said above that it should be even. However when I actually did it, I used the wrong function, which was the function I for some reason didn't foil. I appreciate everyone's help.

 

[MAC5494]

This is going to sound funny, but I typed in the function above wrong. I didn't sleep much last night as you can probably tell.

y=x²(x²-4)²

It is supposed to be y=x²(x-4)²

So I have the same problem. I'll try doing the quadratic formula to see what I get.