Question

How do you find the axis of symmetry, graph and find the maximum or minimum value of the function `F(x)=x^2- 4x -5`?

Answer 1

Answer is:

`x_(symm)=2`

Explanation:

The value of the axis of symmetry in a quadratic polynomial function is:

`x_(symm)=-b/(2a)=-(-4)/(2*1)=2`

Proof

The axis of symmetry in a quadratic polynomial function is between the two roots `x_1` and `x_2`. Therefore, ignoring the y plane, the x value between the two roots is the average `bar(x)` of the two roots:

`bar(x)=(x_1+x_2)/2`

`bar(x)=((-b+sqrt(Δ))/(2a)+(-b-sqrt(Δ))/(2a))/2`

`bar(x)=(-b/(2a)-b/(2a)+sqrt(Δ)/(2a)-sqrt(Δ)/(2a))/2`

`bar(x)=(-2b/(2a)+cancel(sqrt(Δ)/(2a))-cancel(sqrt(Δ)/(2a)))/2`

`bar(x)=(-2b/(2a))/2`

`bar(x)=(-cancel(2)b/(2a))/cancel(2)`

`bar(x)=-b/(2a)`