Question

Function f approximately represents the trajectory of an airplane in an air show, where x is the horizontal distance of the flight, in kilometers. What is the symmetry of the function?

Answer 1

`((3)/(2),102)`

Explanation:

Given

`f(x)=88x^2-264x+300` --- Missing from the question

Required

The symmetry

First, we express f(x) as:
`f(x) = a(x - h)^2 + k`

Where the symmetry is: (h, y) and x = h

Equate f(x) to 0

`88x^2-264x+300 = 0`

Subtract 300 from both sides

`88x^2-264x+300 - 300= 0 - 300`

`88x^2 - 264x= - 300`

Factorize

`88(x^2 - 3x) = -300`

On the left-hand side, the coefficient of x is -3.

Divide by 2 and add the square to both sides.

So, we have:

`88(x^2 - 3x) + ((3)/(2))^2 = -300 + ((3)/(2))^2`

Multiply the new terms by 88 (to make it factorizable)

`88(x^2 - 3x ) +88* ((3)/(2))^2 = -300 + 88 * ((3)/(2))^2`

Factorize

`88(x^2 - 3x + ((3)/(2))^2) = -300 + 88 * ((3)/(2))^2`

The quadratic expression in the bracket is a perfect square. This gives:

`88(x - (3)/(2))^2 = -300 + 88 * ((3)/(2))^2`

`88(x - (3)/(2))^2 = -300 + 88 * (9)/(4)`

`88(x - (3)/(2))^2 = -300 + 22 * 9`

`88(x - (3)/(2))^2 = -102`

Add 102 to both sides

`88(x - (3)/(2))^2 +102= -102 + 102`

`88(x - (3)/(2))^2 +102= 0`

Equate to y

`y = 88(x - (3)/(2))^2 +102`

Recall that:

The symmetry is: (h, y)

By comparison with
`f(x) = a(x - h)^2 + k` and x = h

`h = (3)/(2)`

So:
`x = (3)/(2)`

Substitute
`x = (3)/(2)` in
`y = 88(x - (3)/(2))^2 +102`

`y = 88((3)/(2) - (3)/(2))^2 +102`

`y = 88(0)^2 +102`

`y = 0 +102`

`y = 102`

So, the symmetry is:
`((3)/(2),102)`