Please graph the function and show all work for the table

Question

Given the polynomial: y=x(x+2)(x-3)

complete the table (show all work)

x/ now find the y

-3

-1

1

2

please show all work and graph the function please

Answer 1

y = x(x+2)(x-3)

the solutions are x=0, x=-2, x=3

so the X-intercepts, roots, zeros, are (0,0), (2,0), (3,0)

the y-intercept is ALSO (0,0) becuase y=0 when x=0

so (0,0) serves double duty as BOTH the y-intercept and an x-intercept

SORTS the solutions: -2, 0 , 3

There are 4 intervals if interest:

(-infinity,-2)

(-2,0)

( 0,3 )

(3,infinity)

Next, wisely selects values of x in each of these

intervals, so as to build the sign table

Interval sample text value x f(x)

---------------------------------------------------------

(-infinity,-2) -5 (-5)(-5+2)(-5-3)=

(-5)(-3)(-8) =

-120

(-2,0) -1 (-1)(-1+2)(-1-3)=

(-1)(1)(-4) =4

(0,3) 1 (1)(1+2)(1-3)=

1*3*-2 = -6

(3,infinity) 4 4(4+2)(4-3)=

4*6*1 = 24

So for any x less than -2, the function is negative and the long term

behavior is negative infinity as x grows indefinitely negative.

It is BELOW the x-axis until it hits zero at x=-2.

then it is positive (above the axis), turns around and comes back

down to zero and hits (0,0)

It then is negative , below the x-axis again, turns around and

comes back up to hit (3,0)

finally it is positive forever more for any x>3

you can graph functions at Desmos.

the url is www dot desmos dot com.