Question

graph the reflection (or translation), then find the coordinates of the vertices of each image after the given transformation. 1) reflection across the x-axis 4) reflection across the y-axis 5) translation: 1 unit left and 5 units up 6) translation: 5 units right and 1 unit down

Explanation:

To solve these transformation problems, we'll analyze each one step by step. Let's start with problem 1 (reflection across the x - axis) as an example (assuming the original triangle has vertices, say, \( A(x_1,y_1) \), \( B(x_2,y_2) \), \( C(x_3,y_3) \)):

Problem 1: Reflection across the x - axis

Step 1: Recall the rule for reflection across the x - axis

The rule for reflecting a point \((x,y)\) across the x - axis is \((x,y)\to(x, - y)\).

Step 2: Identify the original coordinates (assume from the graph)

Suppose the original vertices are, for example, \( G(-4,-1) \), \( H(-1,-1) \), \( I(-1,-3) \) (we need to get the exact coordinates from the graph, but the process is the same).

Step 3: Apply the reflection rule

• For \( G(-4,-1) \): After reflection, \( G'(-4,1) \) (since \( y\) - coordinate is negated: \( -(-1)=1 \))
• For \( H(-1,-1) \): After reflection, \( H'(-1,1) \)
• For \( I(-1,-3) \): After reflection, \( I'(-1,3) \)

Problem 4: Reflection across the y - axis

Step 1: Recall the rule for reflection across the y - axis

The rule for reflecting a point \((x,y)\) across the y - axis is \((x,y)\to(-x,y)\).

Step 2: Identify the original coordinates (assume from the graph, say \( M(-5,3) \), \( L(-2,4) \), \( K(-2,0) \))

Step 3: Apply the reflection rule

• For \( M(-5,3) \): After reflection, \( M'(5,3) \) (negate the \( x\) - coordinate: \( -(-5) = 5\))
• For \( L(-2,4) \): After reflection, \( L'(2,4) \)
• For \( K(-2,0) \): After reflection, \( K'(2,0) \)

Problem 5: Translation: 1 unit left and 5 units up

Step 1: Recall the translation rule

The rule for translating a point \((x,y)\) 1 unit left (subtract 1 from \( x\)) and 5 units up (add 5 to \( y\)) is \((x,y)\to(x - 1,y + 5)\).

Step 2: Identify the original coordinates (assume from the graph, say \( O(-4,-3) \), \( J(-1,-3) \), \( D(-1,-1) \))

Step 3: Apply the translation rule

• For \( O(-4,-3) \): \( O'(-4 - 1,-3 + 5)=(-5,2) \)
• For \( J(-1,-3) \): \( J'(-1 - 1,-3 + 5)=(-2,2) \)
• For \( D(-1,-1) \): \( D'(-1 - 1,-1 + 5)=(-2,4) \)

Problem 6: Translation: 5 units right and 1 unit down

Step 1: Recall the translation rule

The rule for translating a point \((x,y)\) 5 units right (add 5 to \( x\)) and 1 unit down (subtract 1 from \( y\)) is \((x,y)\to(x + 5,y - 1)\).

Step 2: Identify the original coordinates (assume from the graph, say \( D(-2,1) \), \( F(-1,0) \), \( E(0,2) \))

Step 3: Apply the translation rule

• For \( D(-2,1) \): \( D'(-2+5,1 - 1)=(3,0) \)
• For \( F(-1,0) \): \( F'(-1 + 5,0 - 1)=(4,-1) \)
• For \( E(0,2) \): \( E'(0 + 5,2 - 1)=(5,1) \)

(Note: The actual coordinates may vary depending on the exact positions of the vertices in the graph. The key is to apply the correct transformation rules: reflection across x - axis: \((x,y)\to(x,-y)\), reflection across y - axis: \((x,y)\to(-x,y)\), translation \( a\) units right/left: \((x,y)\to(x\pm a,y)\), translation \( b\) units up/down: \((x,y)\to(x,y\pm b)\))

If we take problem 1 with original vertices \( G(-4,-1) \), \( H(-1,-1) \), \( I(-1,-3) \):

Answer:

(for problem 1):
The coordinates of the image after reflection across the x - axis are \( G'(-4,1) \), \( H'(-1,1) \), \( I'(-1,3) \)

(For other problems, the answers will be based on the correct application of the respective transformation rules to the actual coordinates of the vertices from the graph)