Question

Describe the graph of the function f(x) = x3 − 18x2 + 101x − 180. Include the y-intercept, x-intercepts, and the shape of the graph.

Can you give me then answer and tell me how you got it and walk me through it.

Answer 1

the y intercept occurs when x = 0

so what do you think the y-intercepts is?

Answer 2

how would I find the Y intercept?

Answer 3

plug in x = 0 and work it out

Answer 4

0^3 - 18(0)^2 + 101(0) - 180 = ?

Answer 5

y=180?

Answer 6

-180

Answer 7

so x=0 and y=-180 how would I figure out the shape of the graph?

Answer 8

well try and find the x intercepts first
they occur when y = 0
so we have the equation
x^3 − 18x^2 + 101x − 180 = 0

Answer 9

so far I got 180=x^3 - 18^2 + 101x

Answer 10

- solve for x

by the rational roots theorem the roots could be factors 0f 180
like +/- 1, +/- 3, +/- 6 , +/-5,+/- 6, +/- 9

so there's a lot to try out!

Answer 11

or we might be able to factor by grouping which would be a lot quicker:-
x3 − 18x2 + 101x − 180

Answer 12

I got 3 answers and I don't know which one is right

x = 9
x = 5
x = 4

Answer 13

nope cant do it that way

Answer 14

how did you get those?

Answer 15

- they might all be right. A cubic equation will have 3 roots.

Answer 16

f(5) = 5^3 - 18(25) + 101(5) - 180 = 0

yes 5 is a root

Answer 17

im so confused, so x would be 5 and y is -180

Answer 18

did you get a calculator or wolfram to do it?

Answer 19

no x = 5 when y = 0

Answer 20

how did you get those 3 roots They are all correct I just checked them on my calc.

Answer 21

I couldn't figure it out so I used a calculator

Answer 22

oh i see
well another way is to to try ouy values such as 3, 5 etc and once you have found one

- say 5- you divide the function by x - 5 using long or synthetic division to get a quadratic which you solve to get the other 2.

Answer 23

to find the shape you can differentiate and find slopes at different values of x
We know it is probably a double loop because there are 3 rational roots.

Answer 24

so the shape is a loop?