Question

How do you use the important points to sketch the graph of `3x^2-8`?

Answer 1

We have:

`y = 3x^2-8`

Let us examine some of the properties of this function:

• This is a polynomial (linear combination of powers of `x` alone). The highest power of `x` is `2`, so it is a polynomial of degree `2`, or a quadratic function.

• The function is quadratic, so will have a `uu` shape or an `nn` . The coefficient of `x` is `3`, which is positive, so it will have a `uu` shape.

• If we look at `f(x)=0` then `3x^2=8 => x^2 = 8/3` leading to two roots, `x=+-sqrt(8/3)`

• If we replace `x` by `-x` then we have

  `f(-x) = 3(-x)^2 -8 = 3(x)^2 -8 = f(x)`
  So, we deduce that `f(x)` is an even function, and symmetrical about `Oy`

• If we put `x=0` then we get:

  `x=0 => f(0) = -8`

Which gives us enough information to sketch the graph, which here is done using the inbuild Socratic graphing functionality:

graph{3x^2-8 [-5, 5, -10, 5]}