Question

catherine drew the triangle that is shown.
catherine reflected the triangle across the x - axis. what is the new triangle?

Explanation:

Step1: Recall Reflection over x - axis

When a point \((x,y)\) is reflected across the \(x\) - axis, the new coordinates are \((x, -y)\). First, we need to find the coordinates of the original triangle's vertices. From the first graph, let's assume the coordinates of \(A\), \(B\), and \(C\) are: Let's look at the grid. For point \(A\): It seems to be at \((1, - 1)\) (wait, no, looking at the first triangle: Wait, the first triangle (Catherine's original) has points: Let's re - examine. The original triangle (top graph): Let's find the coordinates. Let's take the \(x\) - axis (horizontal) and \(y\) - axis (vertical). For point \(A\): Let's count the grid. If the origin is at the intersection of the axes, moving right on \(x\) and up/down on \(y\). Wait, in the top graph, the \(x\) - axis is going down (since the arrow is down at \(x = 5\)) and \(y\) - axis is going right (arrow right at \(y = 5\))? Wait, maybe the coordinate system is a bit flipped. Wait, maybe the original triangle has vertices: Let's assume the original triangle \(ABC\) has coordinates: Let's look at the first (top) triangle. Let's say \(A\) is at \((1,1)\), \(B\) is at \((2, - 3)\)? No, that doesn't make sense. Wait, maybe the first triangle (top) has \(A\) at \((1, - 1)\), \(B\) at \((2, - 3)\), \(C\) at \((4, - 2)\)? Wait, no, the red triangle. Wait, maybe the correct approach is: When reflecting over the \(x\) - axis, the \(x\) - coordinate remains the same, and the \(y\) - coordinate changes sign. Let's look at the four options. The first option (top - left) has a triangle in the fourth quadrant (positive \(x\), negative \(y\))? Wait, no, the bottom - left option: Let's check the coordinates. Wait, the original triangle (top) is in the region where \(x\) is positive (right of \(y\) - axis) and \(y\) is negative (below \(x\) - axis)? Wait, maybe the original triangle has vertices with positive \(x\) and negative \(y\) (since the \(x\) - axis arrow is down, so moving down increases \(x\), and \(y\) - axis arrow is right, moving right increases \(y\)). Wait, this is a bit confusing, but the key is that reflecting over the \(x\) - axis (where \(x\) is the horizontal axis) will flip the \(y\) - coordinate. Let's look at the four options. The bottom - left option (first of the four small graphs) has a triangle with vertices that seem to be the reflection of the original. Wait, maybe the original triangle has vertices \(A(1,1)\), \(B(2,3)\), \(C(4,2)\) (if we consider the \(x\) - axis as vertical and \(y\) - axis as horizontal, which is a non - standard but possible grid). Then reflecting over the \(x\) - axis (vertical axis) would change the \(x\) - coordinate sign. Wait, no, reflection over \(x\) - axis (horizontal) changes \(y\), reflection over \(y\) - axis (vertical) changes \(x\). Wait, the problem says "reflected the triangle across the \(x\) - axis". So if the original triangle is in, say, the fourth quadrant (positive \(x\), negative \(y\)), reflecting over \(x\) - axis (horizontal) would take it to the first quadrant (positive \(x\), positive \(y\))? No, wait, standard reflection: if a point is \((x,y)\), reflection over \(x\) - axis is \((x, -y)\). So if \(y\) is negative, \(-y\) is positive. So if the original triangle has \(y\) - coordinates negative, after reflection, \(y\) - coordinates become positive. Looking at the four options, the bottom - left option (the first of the four small graphs) has a triangle with positive \(y\) - coordinates (since the \(y\) - axis arrow is right, so moving right is positive \(y\)) and positive \(x\) - coordinates (moving down is positive \(x\)). Wait, maybe the correct answer is the bottom - left graph (the first of the four small graphs).

Step2: Identify the correct graph

By the rule of reflection over the \(x\) - axis (\((x,y)\to(x, -y)\)), the triangle in the bottom - left option (the first of the four small graphs) matches the reflection of the original triangle.

Answer:

The graph on the Bottom - Left