Question

How do you graph `y = x^2 - x + 5`?

Answer 1

To graph quadratics, it is usually easier to convert to vertex form, #y = a(x - p)^2 + q

Explanation:

We can convert to vertex form by completing the square.

`y = x^2 - x + 5`

`y = 1(x^2 - x + m) + 5`

`m = (b/2)^2`

`m = (-1/2)^2`

`m = 1/4`

`y = 1(x^2 - x + 1/4 - 1/4) + 5`

`y = 1(x - 1/2)^2 - 1/4+ 5`

`y = (x - 1/2)^2 + 19/4`

Thus, we have our converted equation. Now, let's identify what's what.

The vertex is at `(p, q) -> (1/2, 19/4)`. The vertex is the minimum point in the function.

The value of the parameter a in `y = a(x - p)^2 + q` is 1. Since it's positive, the parabola opens upwards.

The y intercept is at (0, 5). I often recommend finding at least 3 points other than the vertex to make graphing easier. Plug in values for x to find the corresponding y value:

`(2, 7); (3, 11); (-1,7)`

Now, we have enough information to graph:

graph{y = x^2 - x + 5 [-9.375, 10.625, -0.04, 9.96]}

Practice exercises:

Graph the function `y = 1/2x^2 + 3x - 4`

Hopefully this helps!