Question

How do you graph `g(x) = x^2 - 4x + 2`?

Answer 1

See below.

Explanation:

Easiest way is to identify points of interest that help to give the graph some structure.

In general, a polynomial with power `2` is a quadratic, which appears as a parabola (U-shape) on a graph.

Some properties that are useful are intercepts, the axis of symmetry, and the concavity.

We can find the `y`-intercept by setting `x = 0`:

`g(0) = 2`

So we know that the graph intersects the `y`-axis at `(0,2)`.

We can determine an axis of symmetry that equally divides the parabola using the equation `x =-b/(2a)` for an equation of the form `ax^2+bx+c`.

`x = -(-4)/(2(1)) = 4/2 = 2`

So we know the parabola is symmetric about the line `x = 2`.

We can take this further and find the `y` associated with `x =2`, since this will be the minimum, or root of the parabola.

`g(2) = -2`

So we know that the graph has a minimum at `(2,-2)`.

We can determine whether the parabola faces upwards or downwards based on the sign of `a` for an equation of the form `ax^2+bx+c`.

In this case, it's positive. So our parabola has a minimum and opens upward.

This should be sufficient information to sketch the graph rather well. If you want more specificity, you can substitute generic values of `x` into the equation to find the points that lie on the curve.

graph{x^2 - 4x + 2 [-10, 10, -5, 5]}

You should notice that the graph is indeed symmetric about `x = 2` with a minimum of `(2,-2)` and passes through `(0,2)`.

It'd also be easy to deduce that it passes through `(4,2)` as well, since it is symmetric about `x =2` with the point `(0,2)`.