Question

f(x)=-x^2(x-1)(x+3)

Determine the end behavior: find the power function that the graph of f resembles for large values of |x|.
Find the x-and y-intercepts of the graph.
Determine whether the graph crosses or touches the x-axis at each x-intercept.

Answer 1

look at the given polynomial f(x)

whats the degree of f(x) ?

Answer 2

im not given any additional information.. just was given the problem and asked to find that information..

Answer 3

Ahh, actually im asking you a question to see if you know how to find the "degree"

Answer 4

I'm pretty sure i understand how to find the x and y intercepts.. should i treat this as a normal polynomial function and cross multiply?

Answer 5

oh!! i thought would be the exponent

Answer 6

it is indeed a polynomial, but do not multiply yet, i think we can finish the problem with out multiplying

Answer 7

yes, whats the degree of f(x) ?

Answer 8

i mean the degree is the exponent*

Answer 9

2

Answer 10

exponent of what

Answer 11

but then again I'm thinking its just 1

Answer 12

x^2?

Answer 13

f(x)=-x^2(x-1)(x+3)


notice that there are 3 factors here, multiplying them out gives the exponent of leading term as : 2+1+1 = 4

so the degree of f(x) is 4

Answer 14

for part a,
the power function that the graph of f resembles for large values of |x| is \(\large -x^4\)

Answer 15

so as if i was to cross multiply id get an exponent of 2 for the first value and 1 and 1 on the middle and last term?

Answer 16

ok...

Answer 17

Exactly!

\(f(x)=-\color{red}{x}^2(\color{red}{x}-1)(\color{red}{x}+3)\)

its easy to see that \(-\color{Red}{x}^2*\color{Red}{x}*\color{red}{x}\) gives you \(-\color{Red}{x}^{2+1+1} = -\color{red}{x}^4\)

Answer 18

what does finding the degree do for me though?

Answer 19

good question, when you plugin large values of |x|, the function behaves almost same as the power function\(\large -x^4\)

Answer 20

I've been working similar problems and haven't needed that information to find the x and y intercepts

Answer 21

that degree thing helps in answering part a of question :

`Determine the end behavior: find the power function that the graph of f resembles for large values of |x|.`

Answer 22

wow... ok

Answer 23

you can see here, how closely the graph of f(x) matches with the power function \(-x^4\)

https://www.desmos.com/calculator/vj2xhutn6z

Answer 24

i havent been shown how to find the solution to these questions except for finding the x and y intercepts, domain/range, but not the power function.

Answer 25

ok yes!! i see that now.

Answer 26

ok so finding the degree will give me the power function..

Answer 27

Sorry lot of questions... i think I'm just having a hard time understanding what I'm being asked to find.

Answer 28

good, lets find the x intercepts

Answer 29

ok so from what I've understood, to find the x-intercept id want to find the value of the equation by f(x)=0, make the equation equal to 0, and finding the value of x.. thats why i assumed id cross multiply..

Answer 30

but now I'm not seeing how that would work.

Answer 31

thats right!
to find x intercepts, you want to solve \[\large -x^2(x-1)(x+3)=0\]

Answer 32

is this problem similar to the problem f(x)={3+x If x<0 // x^2 if x> not equal to 0....

Answer 33

use zero product property to get :

\(-x^2=0\) or \(x-1= 0\) or \(x+3=0\)

Answer 34

that gives you x intercepts : \(0, 1, -3\)

Answer 35

awesome!! ok so thats what i was thinking..

Answer 36

then id have to find the y intercept?

Answer 37

yes, simply put \(x=0\) in f(x) to get the y intercepts

Answer 38

f(x)=-x^2(x-1)(x+3)

f(0) = ?

Answer 39

-3

Answer 40

wouldn't i substitue all x values with 0?

Answer 41

yes, but -3 is wrong

Answer 42

f(x)=-x^2(x-1)(x+3)

f(0) = -0^2(0-1)(0+3)
= ?

Answer 43

+3

Answer 44

no, try again

Answer 45

what do you get when u multiply something by 0 ?

Answer 46

0

Answer 47

so..

Answer 48

so the answer is 0

Answer 49

cuz im retarded and looked at the problem as if the 0 in front of the parentheses was invisible lol...

Answer 50

haha happens :)

Answer 51

so the y intercept is 0

Answer 52

how do we answer the last part

`Determine whether the graph crosses or touches the x-axis at each x-intercept.`

Answer 53

plot the points and read the graph and how it falls

Answer 54

look at the graph
https://www.desmos.com/calculator/bdlowvla6f

Answer 55

notice that the graph is crossing x axis at x=-3 and x=1
but it is just touching x axis at x=0

Answer 56

ok so it touches the x axis at 0,1,-3 but only crosses at 1 and -3

Answer 57

ooh you answered before i could finish typing.. but that is what i saw

Answer 58

very good!

Answer 59

but then how do know the shape of the graph just by the points.. how do we know how far up it goes and where it lies on the rest of the graph

Answer 60

the graph goes forever on both ends, it sinks down :