Question

the figure in the coordinate plane below is reflected across the y - axis and then reflected across the x - axis.

Explanation:

Step1: Reflect over y - axis

The original figure (smiley) is in the third quadrant (assuming standard coordinate plane, but actually it's below the x - axis and to the left of y - axis, let's say its coordinates are \((-a,-b)\) where \(a>0,b > 0\)). Reflecting over the \(y\) - axis changes the \(x\) - coordinate's sign. So after reflection over \(y\) - axis, the coordinates become \((a,-b)\) (this is in the fourth quadrant).

Step2: Reflect over x - axis

Reflecting the point \((a,-b)\) over the \(x\) - axis changes the \(y\) - coordinate's sign. So the new coordinates are \((a,b)\), which is in the first quadrant. So the final position of the smiley is in the first quadrant (upper - right of the origin).

Answer:

The figure after two reflections (first over \(y\) - axis, then over \(x\) - axis) will be in the first quadrant (with coordinates \((a,b)\) if original was \((-a,-b)\)).