Question

graph the image of δdef after a translation 4 units up.

Explanation:

Step1: Identify original coordinates

First, find the coordinates of \( \triangle DEF \). From the graph:

• \( D \): \( (1, -4) \) (assuming the grid, since \( D \) is 1 unit right on x - axis and 4 units down on y - axis)
• \( E \): \( (8, -4) \) (8 units right on x - axis, 4 units down on y - axis)
• \( F \): \( (-1, 3) \)? Wait, no, looking at the graph, \( F \) is at \( (-1, 3) \)? Wait, no, the blue dot for \( F \) is at \( x=-1, y = 3 \)? Wait, no, the grid: let's re - check. The y - axis: the point \( F \) is at \( x=-1 \), \( y = 3 \)? Wait, no, the original \( F \) is at \( (-1, 3) \)? Wait, no, the user's graph: \( D \) is at \( (1, - 4) \), \( E \) at \( (8, - 4) \), \( F \) at \( (-1, 3) \)? Wait, no, the y - axis has 0 at the center. Wait, the point \( F \) is at \( x=-1 \), \( y = 3 \)? Wait, no, the vertical line (x - coordinate) for \( F \) is \( x=-1 \), and horizontal line (y - coordinate) is \( y = 3 \)? Wait, no, the grid lines: each square is 1 unit. So \( D \) is at \( (1, - 4) \), \( E \) at \( (8, - 4) \), \( F \) at \( (-1, 3) \)? Wait, no, the blue dot for \( F \) is at \( x=-1 \), \( y = 3 \)? Wait, maybe I made a mistake. Let's look again: the y - axis is vertical, x - axis horizontal. The point \( D \) is at \( (1, - 4) \) (1 unit right of y - axis, 4 units below x - axis), \( E \) is at \( (8, - 4) \) (8 units right, 4 units below), \( F \) is at \( (-1, 3) \) (1 unit left, 3 units above x - axis).

Step2: Apply translation (4 units up)

A translation of 4 units up means we add 4 to the y - coordinate of each vertex.

• For \( D(1, - 4) \): New y - coordinate is \( - 4+4 = 0 \), so \( D'=(1, 0) \)
• For \( E(8, - 4) \): New y - coordinate is \( - 4 + 4=0 \), so \( E'=(8, 0) \)
• For \( F(-1, 3) \): New y - coordinate is \( 3 + 4 = 7 \), so \( F'=(-1, 7) \)

Step3: Plot the new points

Now, plot \( D'(1, 0) \), \( E'(8, 0) \), and \( F'(-1, 7) \) on the grid and connect them to form the translated triangle.

Answer:

To graph the image of \( \triangle DEF \) after a translation 4 units up:

1. Find the coordinates of the vertices of \( \triangle DEF \):

• \( D(1, -4) \), \( E(8, -4) \), \( F(-1, 3) \)

1. Apply the translation rule \((x,y)\to(x,y + 4)\):

• \( D'=(1,-4 + 4)=(1,0) \)
• \( E'=(8,-4 + 4)=(8,0) \)
• \( F'=(-1,3 + 4)=(-1,7) \)

1. Plot the points \( D'(1,0) \), \( E'(8,0) \), and \( F'(-1,7) \) on the coordinate plane and draw the triangle connecting these points.