Question

The graph of Q(x)=-(x+3)^2(x-5)^ has x-intercepts at x=-3 and x=5
At and immediately surrounding the point
x = −3 , the graph resembles the graph of
what function?

Answer 1

what is after (x-5)^

Answer 2

nothing

Answer 3

so the ^ should not be there after (x-5) ?

Answer 4

OH!!! its (x-5)^3

Answer 5

We did behavior at end points so far.
Q(x)=-(x+3)^2(x-5)^3
Find the leading term:
(-1)(x^2)(x^3) = -x^5
The ends points behave like -x^3 function, up to the left and down on the right.

But that is not what they are asking here.
If I plot it it looks like an x^2 function near x = -3
Not sure what type of answer is expected. Are there any sample illustration in the book?

Answer 6

it says choose one: y = x y = x^2 y = x^3

Answer 7

Oh, ok. I think they are referring to multiplicity.
The x = -3 root is gotten from the term: (x+3)^2 in the polynomial.
Because of the square the root -3 has a multiplicity of 2. That means near the point x = -3 the function behaves like y = x^2

Answer 8

hmmmm that's a little confusing

Answer 9

could x-3 be considered odd and 5 positive?

Answer 10

To find the roots of Q(x)=-(x+3)^2(x-5)^3 set it to zero and solve for x.
-(x+3)^2 * (x-5)^3 = 0
You will notice x = -3 makes the first factor 0 and x = 5 makes the second factor 0. Either way the product will be 0 and so the roots are x = -3 and x = 5.
But Q(x) can also be written as: Q(x) = -(x+3)(x+3)(x-5)(x_5)(x-5)
So the root x= -3 is repeated twice. So it has a multiplicity of 2
The root x= 5 is repeated three times. So it has a multiplicity of 3

Multiplicity of 2 means the function will behave like x^2 near the root x = -3

Multiplicity of 3 means the function will behave like x^3 near the root x = 5

Answer 11

ohhhh ok

Answer 12

Even multiplicity for a root means the function will behave like x^2 near that root. The graph will touch the x axis and bounce off much like the curve x^2 does at x = 0. The graph will not cut the x axis but touch it.

Odd multiplicity for a root means the function will behave like x^3 near that root. The graph will cut the x axis at that root much like x^3 does at x = 0.

Answer 13

oh okay so x = -3 acts like x^2 because it has an even root =)

Answer 14

The original function Q(x)=-(x+3)^2(x-5)^3 has a root at x = -3. This root has a multiplicity of 2 because of (x+3) * (x+3) in the original function. Since the multiplicity of the root x = -3 is EVEN, the function behave much like x^2 near the root x = -3. The curve will touch the x axis at x = -3 and bounce back on the same side of the x axis. It will not cut the x axis much like x^2 does not cut the x axis at x = 0.

Answer 15

okay ... I am going to post some more and continue to try to understand this ... thank you again

Answer 16

sure. you see a few more examples it will all become very clear.