Question

which statement is true about the graph?
the graph has both point and line symmetry
the graph has line symmetry
the graph has neither point nor line symmetry
the graph has point symmetry

Explanation:

Step1: Recall the definitions

• Line symmetry: A graph has line symmetry if there exists a line (axis of symmetry) such that when the graph is folded along that line, the two halves match exactly.
• Point symmetry: A graph has point symmetry if for every point \((x,y)\) on the graph, there is a corresponding point \((-x,-y)\) on the graph.

Step2: Analyze the given graph

• Check for line symmetry: There is no vertical line \(x = a\) (where \(a\) is a constant) such that folding the graph along \(x=a\) makes the two halves match. For example, if we consider \(x = 0\) (y - axis), the left - hand side of the graph (for \(x<0\)) and the right - hand side (for \(x > 0\)) do not match in terms of shape and position of key points (such as roots and turning points).
• Check for point symmetry: If we take a general point \((x,y)\) on the graph, there is no guarantee that the point \((-x,-y)\) is on the graph. For instance, consider a non - zero \(x\) value where the graph has a root at \(x=-5\) (point \((-5,0)\)), but there is no corresponding point \((5,0)\) in the symmetric (with respect to the origin) sense (the root at \(x = 3\) is not the symmetric counterpart of \(x=-5\) about the origin).

Answer:

The graph has neither point nor line symmetry