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.
Jacinda 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)
The value of h is
and the value of k is
.
The value of h is equal to the horizontal translation of the triangle. By comparing the coordinates of triangle ABC and triangle A'B'C', we can see that the horizontal translation is 3 units to the right. Therefore, the value of h is 3.
The value of k is equal to the vertical translation of the triangle. By comparing the coordinates of triangle ABC and triangle A'B'C', we can see that the vertical translation is 2 units up. Therefore, the value of k is 2.
To determine the values of h and k, we need to analyze the translation of triangle ABC to A'B'C' on the coordinate grid.
Looking at the coordinates of the vertices of both triangles, we can observe that the x-coordinates of the corresponding vertices are the same, indicating a horizontal translation. Similarly, the y-coordinates of corresponding vertices are the same, indicating a vertical translation.
Let's compare the coordinates of the corresponding vertices:
Triangle ABC:
A = (x₁, y₁)
B = (x₂, y₂)
C = (x₃, y₃)
Triangle A'B'C':
A' = (x₁', y₁')
B' = (x₂', y₂')
C' = (x₃', y₃')
Since there is a horizontal translation, the x-coordinates of A'B'C' are obtained by adding h to the x-coordinates of ABC, while the y-coordinates remain the same. Therefore, we have:
x₁' = x₁ + h
x₂' = x₂ + h
x₃' = x₃ + h
Since there is a vertical translation, the y-coordinates of A'B'C' are obtained by adding k to the y-coordinates of ABC, while the x-coordinates remain the same. Therefore, we have:
y₁' = y₁ + k
y₂' = y₂ + k
y₃' = y₃ + k
To find the values of h and k, we need to find the differences between corresponding x-coordinates and y-coordinates:
Difference in x-coordinates:
x₁' - x₁ = x₁ + h - x₁ = h
x₂' - x₂ = x₂ + h - x₂ = h
x₃' - x₃ = x₃ + h - x₃ = h
Difference in y-coordinates:
y₁' - y₁ = y₁ + k - y₁ = k
y₂' - y₂ = y₂ + k - y₂ = k
y₃' - y₃ = y₃ + k - y₃ = k
Therefore, the value of h is equal to the difference in x-coordinates, and the value of k is equal to the difference in y-coordinates.
In conclusion:
The value of h is h
The value of k is k