## anonymous one year ago 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.

1. welshfella

the y intercept occurs when x = 0 so what do you think the y-intercepts is?

2. anonymous

how would I find the Y intercept?

3. welshfella

plug in x = 0 and work it out

4. welshfella

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

5. anonymous

y=180?

6. welshfella

-180

7. anonymous

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

8. welshfella

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

9. anonymous

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

10. welshfella

- 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!

11. welshfella

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

12. anonymous

I got 3 answers and I don't know which one is right x = 9 x = 5 x = 4

13. welshfella

nope cant do it that way

14. welshfella

how did you get those?

15. welshfella

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

16. welshfella

f(5) = 5^3 - 18(25) + 101(5) - 180 = 0 yes 5 is a root

17. anonymous

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

18. welshfella

did you get a calculator or wolfram to do it?

19. welshfella

no x = 5 when y = 0

20. welshfella

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

21. anonymous

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

22. welshfella

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.

23. welshfella

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.

24. anonymous

so the shape is a loop?

25. welshfella

|dw:1441220571153:dw|

26. welshfella

either like that

27. welshfella

or like this |dw:1441220677250:dw|

28. welshfella

f(3) = -12 so it will be the first one

29. welshfella

|dw:1441220810360:dw|

30. anonymous

wow thank you so much!

31. welshfella

yw

32. welshfella

guess you should include the y intercept |dw:1441220974136:dw|

33. welshfella

- in fact i need not have found f(3) because the y intercept was negative anyway

34. welshfella

- it had to be the first shape not the second. - i'd forgotten about the y-intercept