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 xisthe horizontal distance, in feet, from the start of its flight.f(n) = 2x2 - 2x + 20Determine 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 ishorizontally 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 ishorizontally 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 ishorizontally 17 feet and 19 feet away from where it began its flight.​

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 flightStep-by-step explanation:The correct function is[tex]f(x)=(1/2)x^{2}-2x+20[/tex]we have[tex]f(x)=(1/2)x^{2}-2x+20[/tex]This is a vertical parabola open upward (because the leading coefficient is positive)The vertex is a minimumThe equation of the axis of symmetry is equal to the x-coordinate of the vertexsoConvert the quadratic equation in vertex formFactor the leading coefficient 1/2[tex]f(x)=0.5(x^{2}-4x)+20[/tex]Complete the square[tex]f(x)=0.5(x^{2}-4x+4)+20-2[/tex][tex]f(x)=0.5(x^{2}-4x+4)+18[/tex]Rewrite as perfect squares[tex]f(x)=0.5(x-2)^{2}+18[/tex]The vertex is the point (2,18)The x-coordinate of the vertex is 2thereforeThe equation of the axis of symmetry is x=2soThe 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 flightVerifyFor x=1[tex]f(1)=(1/2)(1)^{2}-2(1)+20=18.5\ ft[/tex]For x=3[tex]f(1)=(1/2)(3)^{2}-2(3)+20=18.5\ ft[/tex][tex]f(1)=f(3)[/tex]