Question

PLEASE HELP!
Sketch the graph of the function f(x)=3x^3-24x^2 by a) applying the Leading Coefficient Test, b) finding the zeros of the polynomial, c) plotting sufficient solution points, and d) drawing a continuous curve through the points.

Answer 1

\[3x ^{3} - 24x ^{2}= 3x^ {2}(x-8)\]
\[0=3x^ {2}\] x=0
x-8=0
x=8

End Behavior of graph
When n is odd and an is positive Graph falls to the left and rises to the right
When n is odd and an is negative Graph rises to the left and falls to the right
When n is even and an is positive Graph rises to the left and right
When n is even and an is negative Graph falls to the left and right

So n=3 and is positive so it fall on the left and rise to the right
so it will be something like this

but the leading exponent is a 3 so it will be an s shape like this


to plot the points plug in numbers like 8 0 and -1
so
\[3(-1)^ {2} ((-1)-8)\]= 24 so on the graph put a point on (-1,24)
and 0 will be \[3(0)^ {2} ((0)-8)\]= 0 so on the graph put a point on (0,0)

\[3(8)^ {2} ((8)-8)\]= 0 so one the graph put a point on (8,0)

the the graph will look something like this