Question

graph the image of △def after a reflection over the x-axis.

Explanation:

Step1: Find coordinates of D, E, F

First, we identify the coordinates of the vertices of \(\triangle DEF\). From the graph:

• \(D\): Let's assume the grid has each square as 1 unit. Looking at \(D\), it's at \((3, -7)\) (wait, no, looking again, the y - axis: the point D is at (3, -7)? Wait, no, the y - axis: the bottom part is negative. Wait, let's check the coordinates properly. Let's see the x - coordinate (horizontal) and y - coordinate (vertical).

Wait, the original points: Let's re - examine. Let's take F: from the graph, F is at \((-3, -3)\)? Wait, no, the grid: the x - axis is horizontal, y - axis vertical. Let's look at the positions:

Wait, the point F: x = - 3, y=-3? Wait, no, the blue triangle: F is at (-3, -3)? D is at (3, -7)? E is at (7, -7)? Wait, no, let's check the y - axis. The x - axis is the purple line. Below the x - axis, y is negative. Let's find the coordinates:

• Point F: Let's count the units from the origin (0,0). Moving left 3 units (x=-3) and down 3 units (y = - 3)? Wait, no, the line from F to D and E. Wait, maybe I made a mistake. Let's do it properly.

Wait, the reflection over the x - axis: the rule for reflection over the x - axis is \((x,y)\to(x, - y)\). So first, we need to find the coordinates of D, E, F.

Looking at the graph:

• Point F: Let's see, x - coordinate: - 3 (since it's 3 units to the left of the origin), y - coordinate: - 3? Wait, no, the y - axis: the point F is at y=-3? Wait, the line from F to D: F is at (-3, -3), D is at (3, -7), E is at (7, -7)? Wait, no, D and E are at the same y - level. Let's check the y - coordinate of D and E: they are at y=-7? Wait, the grid lines: from the x - axis (y = 0) down to y=-10. So D is at (3, -7), E is at (7, -7), and F is at (-3, -3).

Step2: Apply reflection rule

The rule for reflection over the x - axis is \((x,y)\to(x, - y)\).

• For point D \((3, - 7)\): Applying the rule, the image \(D'\) will be \((3, -(-7))=(3,7)\)
• For point E \((7, - 7)\): Applying the rule, the image \(E'\) will be \((7, -(-7))=(7,7)\)
• For point F \((-3, - 3)\): Applying the rule, the image \(F'\) will be \((-3, -(-3))=(-3,3)\)

Step3: Plot the reflected points

Now, we plot the points \(D'(3,7)\), \(E'(7,7)\), and \(F'(-3,3)\) on the coordinate plane and connect them to form the reflected triangle \(\triangle D'E'F'\).

Answer:

The image of \(\triangle DEF\) after reflection over the x - axis has vertices at \(D'(3,7)\), \(E'(7,7)\), and \(F'(-3,3)\). (To graph it, plot these three points and connect them.)