Question

sketch a graph that is symmetric to the given graph with respect to the x - axis.

Explanation:

Step1: Recall the rule for x - axis symmetry

If a point \((x,y)\) is on the original graph, then the point \((x, - y)\) is on the graph symmetric about the \(x\) - axis. This means we take each point of the original parabola and reflect it over the \(x\) - axis.

Step2: Identify key points of the original graph

The original graph (a parabola) has a \(y\) - intercept at \((0,3)\) and a vertex at \((- 4,-6)\).

Step3: Reflect key points over the \(x\) - axis

For the point \((0,3)\), its reflection over the \(x\) - axis is \((0,-3)\). For the vertex \((-4,-6)\), its reflection over the \(x\) - axis is \((-4,6)\).

Step4: Sketch the new graph

Using the reflected key points \((0,-3)\) and \((-4,6)\) and the fact that the shape of the parabola (a U - shape) is inverted (since we are reflecting over the \(x\) - axis), we can sketch the graph. The new graph will be a parabola opening upwards (opposite of the original which opened upwards, no, wait original opened upwards? Wait no, original had vertex at \((-4,-6)\) and passed through \((0,3)\), so original was a parabola opening upwards. After reflection over \(x\) - axis, it will open downwards.

Answer:

The graph symmetric about the \(x\) - axis is a parabola with vertex at \((-4,6)\) and passing through \((0,-3)\) (and other reflected points of the original graph), opening downwards.