Question

find the coordinates of each point.
a) point b is the image of h after a translation of 4 units to the left and 6 units up.
the coordinates of point b are ( , ).
b) point c is the image of h after a translation two units to the right, then a reflection using the x - axis.
the coordinates of point c are ( , ).
c) point d is the image of h after a reflection using the y - axis, then a translation two units to the right and three units up.
the coordinates of point d are ( , ).

Explanation:

To solve these problems, we need to know the coordinates of the original point \( H \). Since the coordinates of \( H \) are not provided in the image, we'll assume a general point \( H = (x, y) \) and show the transformation steps for each part.

Part (a): Point \( B \) (Translation 4 units left, 6 units up)

Step 1: Recall translation rules

A translation of \( a \) units left changes the \( x \)-coordinate by \( -a \), and a translation of \( b \) units up changes the \( y \)-coordinate by \( +b \).

Step 2: Apply the translation to \( H = (x, y) \)

• New \( x \)-coordinate: \( x - 4 \) (4 units left)
• New \( y \)-coordinate: \( y + 6 \) (6 units up)

Thus, the coordinates of \( B \) are \( \boldsymbol{(x - 4, y + 6)} \).

Part (b): Point \( C \) (Translation 2 units right, then reflection over \( x \)-axis)

Step 1: Translate \( H = (x, y) \) 2 units right

A translation of 2 units right changes the \( x \)-coordinate by \( +2 \), so the intermediate point is \( (x + 2, y) \).

Step 2: Reflect over the \( x \)-axis

A reflection over the \( x \)-axis changes the \( y \)-coordinate to its opposite: \( (x, y) \to (x, -y) \). Applying this to \( (x + 2, y) \), we get \( (x + 2, -y) \).

Thus, the coordinates of \( C \) are \( \boldsymbol{(x + 2, -y)} \).

Part (c): Point \( D \) (Reflection over \( y \)-axis, then translation 2 units right, 3 units up)

Step 1: Reflect \( H = (x, y) \) over the \( y \)-axis

A reflection over the \( y \)-axis changes the \( x \)-coordinate to its opposite: \( (x, y) \to (-x, y) \).

Step 2: Translate 2 units right

A translation of 2 units right changes the \( x \)-coordinate by \( +2 \): \( -x + 2 \).

Step 3: Translate 3 units up

A translation of 3 units up changes the \( y \)-coordinate by \( +3 \): \( y + 3 \).

Thus, the coordinates of \( D \) are \( \boldsymbol{(-x + 2, y + 3)} \).

Note:

To get numerical coordinates for \( B \), \( C \), or \( D \), substitute the actual coordinates of \( H \) (e.g., if \( H = (h_x, h_y) \), plug \( x = h_x \) and \( y = h_y \) into the formulas above).

Answer:

To solve these problems, we need to know the coordinates of the original point \( H \). Since the coordinates of \( H \) are not provided in the image, we'll assume a general point \( H = (x, y) \) and show the transformation steps for each part.

Part (a): Point \( B \) (Translation 4 units left, 6 units up)

Step 1: Recall translation rules

A translation of \( a \) units left changes the \( x \)-coordinate by \( -a \), and a translation of \( b \) units up changes the \( y \)-coordinate by \( +b \).

Step 2: Apply the translation to \( H = (x, y) \)

• New \( x \)-coordinate: \( x - 4 \) (4 units left)
• New \( y \)-coordinate: \( y + 6 \) (6 units up)

Thus, the coordinates of \( B \) are \( \boldsymbol{(x - 4, y + 6)} \).

Part (b): Point \( C \) (Translation 2 units right, then reflection over \( x \)-axis)

Step 1: Translate \( H = (x, y) \) 2 units right

A translation of 2 units right changes the \( x \)-coordinate by \( +2 \), so the intermediate point is \( (x + 2, y) \).

Step 2: Reflect over the \( x \)-axis

A reflection over the \( x \)-axis changes the \( y \)-coordinate to its opposite: \( (x, y) \to (x, -y) \). Applying this to \( (x + 2, y) \), we get \( (x + 2, -y) \).

Thus, the coordinates of \( C \) are \( \boldsymbol{(x + 2, -y)} \).

Part (c): Point \( D \) (Reflection over \( y \)-axis, then translation 2 units right, 3 units up)

Step 1: Reflect \( H = (x, y) \) over the \( y \)-axis

A reflection over the \( y \)-axis changes the \( x \)-coordinate to its opposite: \( (x, y) \to (-x, y) \).

Step 2: Translate 2 units right

A translation of 2 units right changes the \( x \)-coordinate by \( +2 \): \( -x + 2 \).

Step 3: Translate 3 units up

A translation of 3 units up changes the \( y \)-coordinate by \( +3 \): \( y + 3 \).

Thus, the coordinates of \( D \) are \( \boldsymbol{(-x + 2, y + 3)} \).

Note:

To get numerical coordinates for \( B \), \( C \), or \( D \), substitute the actual coordinates of \( H \) (e.g., if \( H = (h_x, h_y) \), plug \( x = h_x \) and \( y = h_y \) into the formulas above).