# What is the axis of symmetry for f(x) = −2×2 + 20x − 42?

• For a general equation y = ax^2 + bx + c

We need to convert it to vertex form which includes complete factoring.

y = a[ x^2 + (b/a)x + c/a ]

Now examining x^2 + (b/a)x + c/a

If you look at square of bionomial (z + u)^2 = z^2 + 2zu + u^2, and compare it with what we have, you will see the “u” term had to have been b/(2a) so when it’s multiplied by 2z = 2x, you get (b/a)x.

Hence to satisfy the perfect square we must have (b / (2a))^2 = b^2 / (4a^2), but then when we add it we must also subtract it.

x^2 + (b/a)x + b^2/(4a^2) – b^2/(4a^2) + c/a

Now the first 3 terms factor and the other stuff hangs out front.

(x + b/(2a))^2 – b^2/(4a^2) + c/a

And now finally returning to

y = a[ (x + b/(2a))^2 – b^2/(4a^2) + c/a ]

y = a[x + b/(2a)]^2 – b^2/(4a)+c

So suppose a is positive, then the parabola will open up. Where is the expression of lowest value? Obviously when x + b/(2a) = 0, meaning x = -b / (2a) which is where the vertex is and that is the axis of symmetry.

So now you can find the axis of symmetry of any parabola of the given form by just doing -0.5b/a.

• You have some notation problems, but it looks like you are trying to define a quadratic function in expanded form:

f(x) = ax² + bx + c

In that case, the graph of the equation y = f(x) has this axis of symmetry:

x = -b/(2a)