Before we get to the question, use the picture on board above to revise that:

• the first derivate of a cubic function is a parabola
• and the turning points of the cubic function are x-intercepts of the parabola
• the point of inflection shares the x-coordinate of the parabola turning point

9.1

Straightforward and very fast to the answer, sub zero on x to get y intercept

9.2

And sub zero on y to get the x intercepts.

To solve for x we use factor theorem, in this case 1 is a factor and x-1 is our factor bracket, but that gets us to incorrect three identical factor brackets of x-1

I had to inspect and redo this one, I used -1 as a factor and x+1 as my factor bracket, and the used synthetic division to get values of a b and c as in the white board bellow

Synthetic division:

• write the coefficient of the first three terms of the cubic polynomial and the constant on a number line
• drop the first number below the L shape number line and multiply by the factor
• add the answer to the next number and repeat until the last number gives zero

from here we can the factorise the quadratic equation as in the white board

9.3

Coordinates of the turning point are simply second derivative equal to zero

And then we solve for x, sub x on original equation of the cubic graph to find y values of the TP

9.4

Make a dot of all coordinates we found so far, and connect them to make a positive cubic graph.

i.e. starts with maximum turning point and goes to minimum TP

and vice versa if the cubic function was negative

9.5

Remember the first task to revise relationship of cubic function and the parabola? It is essential for you to understand this question

The second derivative is a parabolic function and is positive as drawn in the first white board.

As per inspection, the parabola is less than zero, below x axis, and that is from x intercept to the next x intercept of the parabola

Which is in essence, the turning points of the cubic function

Answer: