Question is in description

Question

Consider the following claim: If the point (2+d,y) is on the graph of the function f(x)=x(x-4), then point (2-d,y) is also on the graph.

•Use algebra to show that the claim is true.

•What is the relationship between the line x=2 and the graph of f(x)?

Answer 1

f(x) = x(x-4) = x^2 -4x = x^2-4x + 4 - 4 = (x-2)^2 -4 = (x-h)^2 -k where (h,k) is the vertex (2.-4)

the function is a parabola with vertex at (2,-4)

f(2+d)= f(2-d) = y

because the line of symmetry is x=2

put a mirror on the line x=2 and the left side of the parabola is a mirror image of the right side

go d units to the left of x=2 and the y value on the parabola is the same was when you go d units to the right

plug 2-d into the parabola's equation, f(x) = (x-2)^ -4

f(2-d) = (2-d-2)^2-4 = d^2-4

plug 2+d into the parabola's equation

f(2+d) = (2+d-2)^2-4 = d^2-4, the same y value as f(2-d)

the relationship of x=2 to the graph of f(x) is that x=2 is the function, f(x)'s line of symmetry