1) The function is 3(x + 2)³ - 3
2) The end behaviour is the limits when x approaches +/- infinity.
3) Since the polynomial is of odd degree you can predict that the ends head off in opposite direction. The limits confirm that.
4) The limit when x approaches negative infinity is negative infinity, then the left end of the function heads off downward (toward - ∞).
5) The limit when x approaches positive infinity is positivie infinity, then the right end of the function heads off upward (toward + ∞).
6) To graph the function it is important to determine:
- x-intercepts
- y-intercepts
- critical points: local maxima, local minima, and inflection points.
7) x-intercepts ⇒ y = 0
⇒ 3(x + 2)³ - 3 = 0 ⇒ (x + 2)³ - 1 = 0
⇒ (x + 2)³ = -1 ⇒ x + 2 = 1 ⇒ x = - 1
8) y-intercepts ⇒ x = 0
y = 3(x + 2)³ - 3 = 3(0 + 2)³ - 3 = 0 - 3×8 - 3 = 24 - 3 = 21
9) Critical points ⇒ first derivative = 0
i) dy / dx = 9(x + 2)² = 0
⇒ x + 2 = 0 ⇒ x = - 2
ii) second derivative: to determine where x = - 2 is a local maximum, a local minimum, or an inflection point.
y'' = 18 (x + 2); x = - 2 ⇒ y'' = 0 ⇒ inflection point.
Then the function does not have local minimum nor maximum, but an inflection point at x = -2.
Using all that information you can graph the function, and I attache the figure with the graph.
