Question

Question #bc867

Answer 1

Factor it to find the zeros at `x=-1,3`, and find the vertex at `(1,-8)`. Plot those points, and there's your parabola! graph{2x^2-4x-6 [-20, 20, -10, 10]}

Explanation:

Factoring is the fastest way to find the zeros of this function.
Factor out the GCF first: `2x^2-4x-6 = 2(x^2-2x-3)`.
Factor the second part: `x^2-2x-3 = (x-3)(x+1)`.
Put it all together: `g(x)=2(x-3)(x+1)`

The solutions/zeros/roots are where the function crosses the
x-axis, which means y=0 at those points. Set `g(x)=0`:
`2(x-3)(x+1)=0`.
There are two ways the product of these three terms equals zero:
If `(x-3)=0` or if `(x+1)=0`, then `g(x)=0`
Solve the two equations to get `x=3` and `x=-1`.
These are the places that the function crosses the x-axis.

Find the vertex:
The x-coordinate of the vertex of a parabola is always at `-b/(2a)`. Plug that number back into the original function to get the corresponding y-coordinate.
`-b/(2a)=(-(-4))/(2(2))=4/4=1` is the x-coordinate of the vertex.
Find the y-coordinate: `g(1)=2(1)^2-4(1)-6=2-4-6=-8`
The vertex is at `(1,-8)`.
Plot the vertex, plot the zeros, and draw your parabola!