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

Question

Consider the following claim: 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.

 

Enter your work, your answer, and your justification in the box provided.

Answer 1

Claim: 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.

Proof. [-:)]. Point (2 + d, y) is on the graph of f(x) = x(x - 4) means that

y=(2+d)(2+d-4)=(d+2)(d-2)=(2-d)(-(2+d))=(2-d)(2-d- 4)=f(2-d), what is reuqed to prove.

Line x=2 and f(x)=x(x-4) .

Let's rewrite f(x) as f(x)==(x-2+2)(x-2-2)=(x-2)² -4. This the vertex form of f(x). and when x=2, then f(2)=-4. So x=2 is the x coordinate of the vertex of f(x) and a point where f(x) reaches it minimum (-4).