Question

which graph shows line symmetry about the y-axis?

Explanation:

Step1: Recall symmetry about y-axis

A graph has line symmetry about the \( y \)-axis if for every point \((x, y)\) on the graph, the point \((-x, y)\) is also on the graph. This means the graph is a mirror image across the \( y \)-axis.

Step2: Analyze each graph

• First graph ( \( k(x) \)): The left side and right side do not mirror each other across the \( y \)-axis (e.g., the vertex is not symmetric about \( y \)-axis, and the shape on left and right of \( y \)-axis differs).
• Second graph ( \( f(x) \)): The graph is not symmetric about \( y \)-axis (it is more like a function that is not mirrored, e.g., the left side near \( x=-1 \) and right side near \( x = 1 \) do not match).
• Third graph ( \( g(x) \)): For every point \((x, y)\) on the graph, the point \((-x, y)\) appears to be on the graph. For example, the peak at \( x = 4 \) has a corresponding peak at \( x=-4 \), and the valley at \( x = 0 \) is on the \( y \)-axis, so it is symmetric about \( y \)-axis.
• Fourth graph ( \( h(x) \)): The left side (e.g., \( x=-1 \)) and right side (e.g., \( x = 1 \)) do not match in terms of \( y \)-values, so not symmetric about \( y \)-axis.

Answer:

The graph of \( g(x) \) (the third graph) shows line symmetry about the \( y \)-axis.