Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `f(x)=x ^2 + 3x-10`?

Answer 1

Complete the square

Explanation:

After completing the square

`f(x) = x^2 + 3x - 10 -= frac{(2x+3)^2 - 49}{4}`

From here, we can see that

`f(-3/2-x) = frac{4x^2 - 49}{4} = f(-3/2+x)`

Therefore the axis of symmetry is `x=-3/2`.

We also know that `(2x+3)^2>=0`. The minimum for `f` corresponds to the value of `x` which equality holds, i.e. `(2x+3)^2=0`.

Solving it gives `x=-3/2`, which is unsurprising if you already know that the minimum/maximum of a parabola lies on its axis of symmetry.

`f(-3/2) = -49/4` is the minimum.