Question

A bird takes flight from a branch in a tree. The given function represents the flight of the bird, where f(x) is the height of the bird, in feet, and xis
the horizontal distance, in feet, from the start of its flight.
f(n) = 2x2 - 2x + 20
Determine the symmetry of the function.
A. The flight of the bird is symmetric about the line x = 4 feet, which indicates that the bird is at the same height when it is
horizontally 3 feet and 5 feet from where it began its flight.

B.The flight of the bird is not symmetric.
C. The flight of the bird is symmetric about the line x = 2 feet, which indicates that the bird is at the same height when it is
horizontally 1 foot and 3 feet away from where it began its flight.
D. The flight of the bird is symmetric about the line x = 18 feet, which indicates that the bird is at the same height when it is
horizontally 17 feet and 19 feet away from where it began its flight.​

Answer 1

Answer:

Option C. The flight of the bird is symmetric about the line x = 2 feet, which indicates that the bird is at the same height when it is  horizontally 1 foot and 3 feet away from where it began its flight

Step-by-step explanation:

The correct function is

$f(x)=(1/2)x^{2}-2x+20$

we have

$f(x)=(1/2)x^{2}-2x+20$

This is a vertical parabola open upward (because the leading coefficient is positive)

The vertex is a minimum

The equation of the axis of symmetry is equal to the x-coordinate of the vertex

so

Convert the quadratic equation in vertex form

Factor the leading coefficient 1/2

$f(x)=0.5(x^{2}-4x)+20$

Complete the square

$f(x)=0.5(x^{2}-4x+4)+20-2$

$f(x)=0.5(x^{2}-4x+4)+18$

Rewrite as perfect squares

$f(x)=0.5(x-2)^{2}+18$

The vertex is the point (2,18)

The x-coordinate of the vertex is 2

therefore

The equation of the axis of symmetry is x=2

so

The flight of the bird is symmetric about the line x = 2 feet, which indicates that the bird is at the same height when it is  horizontally 1 foot and 3 feet away from where it began its flight

Verify

For x=1

$f(1)=(1/2)(1)^{2}-2(1)+20=18.5\ ft$

For x=3

$f(1)=(1/2)(3)^{2}-2(3)+20=18.5\ ft$

$f(1)=f(3)$