We first identify that `a = -1`, `b = 1` and `c = 6`

Step (a) Since `a = -1`, `a < 0` hence the function is a parabola with a maximum point and it opens downwards (n-shaped)

Step (b) The x co-ordinate of the maximum point is

`x=-b/(2a)=(-(1))/(2(-1))=(-1)/-2=1/2`

So

`y=-(1/2)^2+1/2+6=6 1/4`

Step (c) The y-intercept is the point `(0, c) = (0, 6)`

Sometimes, we need some more points to get a better sketch of the parabola.

Two points we can also find are the x-intercepts ie. points where the function cuts the x axis (where `y = 0`).

To find the x intercepts, we let `y = 0` and we get `-x^2 + x + 6 = 0`

or `x^2 - x - 6 = 0`

Factoring `(x - 3) (x + 2) = 0`

Solving `x - 3 = 0` or `x + 2 = 0`

So `x = 3` or `x = -2`.

Hence, the x intercepts are `(3, 0)` and `(-2, 0)`.

Here is our sketch:

123456-1-2-3-4510-5-10xy(-2,0)(0,6)(0.5,6.25)(3,0)Open image in a new page

`y = -x^2 + x + 6`, showing maximum point and intersections with axes