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Question

A coordinate plane shows two points upper A and upper A prime plotted in quadrant 2 and quadrant 1, respectively. Both axes range from negative 5 to 5 in increments of 1.

On this coordinate plane, point A is translated into point A′. Which functions give the two translations?

(1 point)
Responses

f(x)=x+6 and g(y)=y+3
f left parenthesis x right parenthesis equals x plus 6 and g left parenthesis y right parenthesis equals y plus 3

f(x)=x+2 and g(y)=y+5
f left parenthesis x right parenthesis equals x plus 2 and g left parenthesis y right parenthesis equals y plus 5

f(x)=x−5 and g(y)=y−2
f left parenthesis x right parenthesis equals x minus 5 and g left parenthesis y right parenthesis equals y minus 2

f(x)=x+5 and g(y)=y+2
f left parenthesis x right parenthesis equals x plus 5 and g left parenthesis y right parenthesis equals y plus 2
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Answer 1

To answer the question about the translations from point A to point A′, we need to determine the changes in both the x-coordinate and y-coordinate.

1. Identify Points: The coordinates of point A in quadrant 2 will be of the form (-x, y) and will be negative in the x-direction. The coordinates of point A′ in quadrant 1 will be positive in both the x and y-directions.

2. Determine the Transformation: The translation from A to A' involves determining how much the x-coordinate and y-coordinate change. For example, if:

   • A has coordinates A(-x_A, y_A)
   • A' has coordinates A'(x_A', y_A')

   The transformations can be described as:

   • f(x): x-coordinate transformation (change in x)
   • g(y): y-coordinate transformation (change in y)

3. Analyze Function Options: Each option provides a pair of functions representing changes in the x and y coordinates.

   • We can investigate the differences based on the proposed functions, noting that increasing x indicates a rightward translation and decreasing y indicates a downward translation.

Given the options:

1. \(f(x) = x + 6\) and \(g(y) = y + 3\) implies a shift right and up.
2. \(f(x) = x + 2\) and \(g(y) = y + 5\) implies a shift right and up.
3. \(f(x) = x - 5\) and \(g(y) = y - 2\) implies a shift left and down.
4. \(f(x) = x + 5\) and \(g(y) = y + 2\) implies a shift right and up.

Without the actual coordinates of points A and A′, the specific transformations can’t be conclusively determined. However, if point A is in quadrant II and point A′ is in quadrant I, the x-coordinate must shift to the right (positive) and the y-coordinate must also shift up (positive).

Therefore, the most likely candidates for the functions could be those which involve positive shifts for both x and y. As a result, one of the first two options (that both increase coordinates) would be plausible.

If the details of the coordinates were clearer, one could definitively select the appropriate option. Based on typical transformations regarding quadrants:

Final Selected Responses:

• Option 1 or Option 2 would be the candidates, depending on the specific translation amounts. Ensure to check point A and A′'s coordinates in the context given.