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Use the image to answer the question.

A coordinate plane shows two plotted triangles upper A upper B upper C, and upper A prime upper B prime upper C prime, in the same orientation. Both axes range from negative 7 to 7 in increments of 1.

Dianna draws triangle ABC
and translates it along the coordinate grid to produce triangle A′B′C′
. Then, she determines the functions f(x)=x+h
and g(y)=y+k
that describe the translation. What are the values of h and k in these functions?

(1 point)
Responses

h=2
and k=−5
h equals 2 and k equals negative 5

h=−8
and k=−5
h equals negative 8 and k equals negative 5

h=5
and k=−8
h equals 5 and k equals negative 8

h=8
and k=−5

1 answer

To determine the values of \( h \) and \( k \) for the functions \( f(x) = x + h \) and \( g(y) = y + k \), we need to figure out the horizontal and vertical translations of triangle \( ABC \) to triangle \( A'B'C' \).

  1. Horizontal Translation (h): This is determined by how much the triangle has moved left or right. If the triangle has moved to the right, \( h \) is positive. If it has moved to the left, \( h \) is negative.

  2. Vertical Translation (k): This is determined by how much the triangle has moved up or down. If the triangle has moved up, \( k \) is positive. If it has moved down, \( k \) is negative.

In this situation, since I cannot see the image or the coordinates directly, I can't provide the exact values of \( h \) and \( k \). However, you can find them by comparing the coordinates of the original triangle \( ABC \) with the translated triangle \( A'B'C' \).

For example, if point \( A \) has coordinates \( (x_1, y_1) \) and point \( A' \) has coordinates \( (x_1', y_1') \), then:

  • \( h = x_1' - x_1 \)
  • \( k = y_1' - y_1 \)

Using this method for any two corresponding points in the triangles will yield the values of \( h \) and \( k \).

If you know the coordinates, please share them for a more definitive answer, or you can apply the above method to find \( h \) and \( k \). If the triangle was translated by 2 units to the right and 5 units downward, then the answer would be: \( h = 2 \) and \( k = -5 \).