Question

assume that the following graph has the given type of symmetry.
x-axis symmetry
complete the graph of the equation.

Explanation:

Step1: Recall x - axis symmetry rule

For a graph with x - axis symmetry, if a point \((x,y)\) is on the graph, then the point \((x, - y)\) is also on the graph.

Step2: Analyze the given graph

The original graph (the upper part) has a point at \((5,0)\) (the vertex) and a curve in the first quadrant (where \(y\geq0\)). To apply x - axis symmetry, we need to reflect each point \((x,y)\) on the original graph over the x - axis, which means for each point, we keep the \(x\) - coordinate the same and change the sign of the \(y\) - coordinate.

Step3: Identify the correct graph

• The first option (the one with the "U" - shaped curve above the x - axis) is symmetric about the y - axis, not the x - axis.
• The second option (the curve going from the third quadrant to the first quadrant) is symmetric about the origin, not the x - axis.
• The third option: For the original graph (upper part), when we reflect over the x - axis, the part below the x - axis should be the mirror image of the part above the x - axis with respect to the x - axis. So if we have a curve in the first quadrant (above x - axis) with vertex at \((5,0)\), the reflected curve should be in the fourth quadrant (below x - axis) with the same \(x\) - coordinates and \(y\) - coordinates negated. This matches the third graph.

Answer:

The third graph (the one with the curve opening to the right, with one part above the x - axis and one part below the x - axis, symmetric about the x - axis)