If the point (2 + d, y) is on the graph of the function f(x) = x(x – 4), then the 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)? Justify your reasoning.
If f(x) = x(x  4)
When x = (2 + d) f(x) = (2 + d)((2 + d  4) {substituted (2 + d) in for x, into f(x) = x(x  4)} f(x) = (2 + d)(2 + d) {combined like terms in parentheses} When x = (2  d) f(x) = (2  d)(2  d  4) {substituted (2  d), in for x, into f(x) = x(x  4)} f(x) = (2  d)(2 + d) {combined like terms in parentheses} With f(x) being the same, whether x is (2 + d) or (2  d), this proves the claim is true. The line x = 2 is the axis of symmetry of the parabola. Using the graphing calculator, you can see x = 2 is the vertical line that passes through the vertex of the parabola f(x) = x(x  4).
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