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

1 Answer
Jan 23, 2016

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.