Question

example 1 determine whether each function has a line of symmetry. explain. state the equation of the line of symmetry, if one exists. a. b.

Explanation:

Step1: Recall line - symmetry concept

A function has a line of symmetry if the graph can be folded along a line such that the two halves match exactly.

Step2: Analyze function \(f(x)\)

The function \(y = f(x)\) is a linear function. A non - vertical and non - horizontal linear function has no line of symmetry. If we try to fold the graph of \(y = f(x)\) along any line, the two halves will not match.

Step3: Analyze function \(g(x)\)

The function \(y = g(x)\) is a periodic function. It has a line of symmetry. The graph of \(y = g(x)\) is symmetric about the vertical line \(x = 1.5\) (by observing the graph, the part of the graph to the left of \(x = 1.5\) is a mirror image of the part to the right of \(x = 1.5\)).

Answer:

a. The function \(f(x)\) has no line of symmetry because a non - vertical and non - horizontal linear function does not have a line along which it can be folded to make the two halves match.
b. The function \(g(x)\) has a line of symmetry. The equation of the line of symmetry is \(x = 1.5\).