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

Answer:

$(\frac{3}{2},102)$

Step-by-step 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:

$88(x^2 - 3x) + (\frac{3}{2})^2 = -300 + (\frac{3}{2})^2$

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

$88(x^2 - 3x ) +88* (\frac{3}{2})^2 = -300 + 88 * (\frac{3}{2})^2$

Factorize

$88(x^2 - 3x + (\frac{3}{2})^2) = -300 + 88 * (\frac{3}{2})^2$

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

$88(x - \frac{3}{2})^2 = -300 + 88 * (\frac{3}{2})^2$

$88(x - \frac{3}{2})^2 = -300 + 88 * \frac{9}{4}$

$88(x - \frac{3}{2})^2 = -300 + 22 * 9$

$88(x - \frac{3}{2})^2 = -102$

Add 102 to both sides

$88(x - \frac{3}{2})^2 +102= -102 + 102$

$88(x - \frac{3}{2})^2 +102= 0$

Equate to y

$y = 88(x - \frac{3}{2})^2 +102$

Recall that:

The symmetry is: (h, y)

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

$h = \frac{3}{2}$

So: $x = \frac{3}{2}$

Substitute $x = \frac{3}{2}$ in $y = 88(x - \frac{3}{2})^2 +102$

$y = 88(\frac{3}{2} - \frac{3}{2})^2 +102$

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

$y = 0 +102$

$y = 102$

So, the symmetry is: $(\frac{3}{2},102)$