Question

draw the image of $\triangle abc$ under a translation by 1 unit to the left and 5 units up.

Explanation:

Step1: Identify coordinates of \( \triangle ABC \)

First, we find the coordinates of points \( A \), \( B \), and \( C \) from the graph.

• Point \( A \): \( (0, -5) \)
• Point \( B \): \( (2, 2) \)
• Point \( C \): \( (7, 0) \) (Wait, no, looking at the blue and teal points, let's re - check. Wait, the blue triangle: Let's see the blue points. Wait, maybe I misidentified. Let's look again. The original triangle \( \triangle ABC \): Let's find the coordinates correctly. Let's take the blue - colored (or the ones forming the triangle) points. Wait, point \( A \) is at \( (0, - 5) \), point \( B \) is at \( (2, 2) \), point \( C \) is at \( (7, 0) \)? No, maybe the blue triangle has vertices: Let's check the grid. Wait, another approach: Translation rule for a point \( (x,y) \) under translation \( h \) units left (so \( x\to x - h \)) and \( k \) units up (so \( y\to y + k \)). Here, \( h = 1 \) (left 1 unit) and \( k = 5 \) (up 5 units). So the translation rule is \( (x,y)\to(x - 1,y + 5) \).

Step2: Translate point \( A \)

Point \( A \): Let's assume the correct coordinates of \( A \) is \( (0, - 5) \). Applying the translation: \( x'=0 - 1=-1 \), \( y'=-5 + 5 = 0 \). So \( A'=(-1,0) \).

Step3: Translate point \( B \)

Point \( B \): Let's say \( B=(2,2) \). Then \( x'=2 - 1 = 1 \), \( y'=2 + 5 = 7 \). So \( B'=(1,7) \).

Step4: Translate point \( C \)

Point \( C \): Let's say \( C=(7,0) \). Then \( x'=7 - 1 = 6 \), \( y'=0 + 5 = 5 \). So \( C'=(6,5) \).

Step5: Draw the translated triangle

After finding the translated points \( A'(-1,0) \), \( B'(1,7) \), and \( C'(6,5) \), we plot these points on the grid and connect them to form the image of \( \triangle ABC \) after translation.

(Note: If the initial coordinates of \( A \), \( B \), \( C \) were misidentified, we can re - identify. For example, if point \( A \) is at \( (0, - 5) \), point \( B \) at \( (2,2) \), point \( C \) at \( (7,0) \), the translation gives the new points as above. If the original triangle has different coordinates, the process is the same: apply \( (x - 1,y + 5) \) to each vertex.)

Answer:

To draw the image of \( \triangle ABC \) under translation 1 unit left and 5 units up, follow these steps:

1. Determine the coordinates of vertices \( A \), \( B \), and \( C \) of \( \triangle ABC \) from the graph. For example, if \( A=(0, - 5) \), \( B=(2,2) \), \( C=(7,0) \):

• Translate \( A \): \( (0 - 1,-5 + 5)=(-1,0) \)
• Translate \( B \): \( (2 - 1,2 + 5)=(1,7) \)
• Translate \( C \): \( (7 - 1,0 + 5)=(6,5) \)

1. Plot the points \( (-1,0) \), \( (1,7) \), and \( (6,5) \) on the coordinate grid and connect them to form the translated triangle \( \triangle A'B'C' \).