Question

How do you graph `y=x^3-4x^2+x+6`?

Answer 1

Factorise to find x-intercepts

Explanation:

Look at the constant at the end (6) and use the factors of that number to find one of your roots. For example, if you put x = 2, that gives you 8-16+2+6 which = 0 therefore (x-2) is a factor.

You now know that the equation is a combination of (x-2) and, because it is a cubic graph, a quadratic is left over.

`y=(x-2)(ax^2+bx+c)`

Now as you look at the coefficients of each term, the coeff of `x^3` is 1 therefore a must be 1

To find the coeff of `x^2`, it is -4 which must equal b - 2a (a combination of `bx` x `x` and `ax` times `-2`)

Therefore `b = -2`

Finally, you need the c which is 6 and is `-2c` from the expansion so `c = -3`

Altogether this gives us `y=(x-2)(x^2-2c-3)`

The quadratic can be factorised to give:

`y=(x-3)(x+1)`

to give `y=(x-3)(x-2)(x+1)`

These three brackets give us our three x intercepts (-1, 2 and 3) and when x = 0, y = 6 which is the y intercept!!

Don't forget cubic graphs are like sideways 's'