Question

practice
1 write a function to describe each translation. each solid figure is a pre - image, and each
dashed figure is an image.
(a) image of a rectangle translation with points a and b
(b) image of a trapezoid translation with points a and b
(c) image of a hexagon translation with points p and q
(d) image of a triangle translation with points p and q

Explanation:

Step1: Analyze Translation Vector (a)

For part (a), observe the vector from point \( A \) to \( B \). Let's assume the coordinate system (though visual, we can infer horizontal and vertical shifts). If \( A \) to \( B \) is, say, \( (h, k) \), the translation function for a point \( (x, y) \) is \( (x + h, y + k) \). From the diagram, the solid figure (pre - image) to dashed (image) has the same vector as \( \overrightarrow{AB} \). So we determine the horizontal and vertical components of \( \overrightarrow{AB} \).

Step2: Analyze Translation Vector (b)

For part (b), the vector is from \( B \) to \( A \). Let the coordinates of \( B \) and \( A \) give the shift. The translation function will be based on the change in \( x \) and \( y \) from \( B \) to \( A \), so for a point \( (x, y) \), it's \( (x + \Delta x, y + \Delta y) \) where \( \Delta x \) and \( \Delta y \) are the differences in \( x \) and \( y \) coordinates of \( A - B \).

Step3: Analyze Translation Vector (c)

For part (c), the vector is from \( Q \) to \( P \). The pre - image (dashed) to image (solid) has a translation vector equal to \( \overrightarrow{QP} \). We find the horizontal and vertical shifts from \( Q \) to \( P \) and apply to a point \( (x, y) \) as \( (x + h, y + k) \) where \( h \) and \( k \) are the components of \( \overrightarrow{QP} \).

Step4: Analyze Translation Vector (d)

For part (d), the vector is from \( Q \) to \( P \) (or vice - versa, need to check pre - image and image). The pre - image is dashed, image is solid. The translation vector is determined by the shift from the pre - image to image, which is related to the vector between \( Q \) and \( P \). So for a point \( (x, y) \), the translation function is \( (x + \Delta x, y + \Delta y) \) based on the coordinate change from pre - image to image.

(Note: Since the exact coordinates are not given, we can describe the general method. For a specific example, if in part (a), suppose \( A=(x_1,y_1) \) and \( B=(x_1 + h,y_1 + k) \), then the translation function is \( T(x,y)=(x + h,y + k) \). Similarly for other parts, we identify the vector between the corresponding points (like pre - image to image vertices or the given points) and write the function as \( (x + \Delta x,y + \Delta y) \) where \( \Delta x \) and \( \Delta y \) are the horizontal and vertical shifts respectively.)

Answer:

(For a more specific answer, we need exact coordinate information. But in general, for each part:

a) Let the vector from \( A \) to \( B \) be \( (h,k) \), then the translation function is \( T(x,y)=(x + h,y + k) \)

b) Let the vector from \( B \) to \( A \) be \( (h,k) \), then the translation function is \( T(x,y)=(x + h,y + k) \)

c) Let the vector from \( Q \) to \( P \) be \( (h,k) \), then the translation function is \( T(x,y)=(x + h,y + k) \)

d) Let the vector from the pre - image (dashed) to image (solid) be \( (h,k) \) (related to \( \overrightarrow{QP} \) or \( \overrightarrow{PQ} \)), then the translation function is \( T(x,y)=(x + h,y + k) \)

If we assume some simple shifts (e.g., in (a) if \( \overrightarrow{AB}=(2,1) \), then \( T(x,y)=(x + 2,y + 1) \); but without exact coordinates, we provide the general form. However, since the problem is about writing the function, we can describe the process as above.)