graph the image of each figure under a translation by the given vector.
pentagon DEFGH with vertices D(0, 5), E(0, 7), F(−2, 6), G(−3, 4), H(−2, 3); g = < 4, − 2 >
I understand how to graph it, but I don't understand the purpose of (g = < 4, − 2 >) or how to apply it.
I have never seen this notation either but suspect it means move all the points right 4 and down 2
like D(0,5) becomes (4 , 3) and E becomes (4, 5)
In this question, the vector g = <4, -2> represents a translation. It indicates how the shape is shifted or moved in the coordinate plane.
To apply the translation, you need to add the vector g to the coordinates of each vertex of the original pentagon. The vector g has two components: 4 for the x-axis and -2 for the y-axis.
Let's take an example to see how this works:
Original pentagon DEFGH has the following vertices:
D(0, 5), E(0, 7), F(−2, 6), G(−3, 4), H(−2, 3)
To apply the translation by g = <4, -2>, you add 4 to the x-coordinates and subtract 2 from the y-coordinates of each vertex:
New Vertex for D: (0 + 4, 5 - 2) = (4, 3)
New Vertex for E: (0 + 4, 7 - 2) = (4, 5)
New Vertex for F: (−2 + 4, 6 - 2) = (2, 4)
New Vertex for G: (−3 + 4, 4 - 2) = (1, 2)
New Vertex for H: (−2 + 4, 3 - 2) = (2, 1)
Now you have the coordinates of the image of each vertex under the given translation. Plot these new vertices on the coordinate plane to obtain the image of the pentagon after the translation.
In this question, the vector g = <4, -2> represents the translation vector. It provides the information on how to move each point of the original figure to its new position.
To apply the translation, you need to add the components of the vector g to the corresponding coordinates of each vertex of the pentagon. Here's how you can do it step-by-step:
1. Start with the original pentagon, which has vertices as follows:
D(0, 5), E(0, 7), F(-2, 6), G(-3, 4), H(-2, 3).
2. To translate each point, add the components of the vector g = <4, -2> to the respective x and y coordinates:
D' = D + g = (0 + 4, 5 - 2) = (4, 3)
E' = E + g = (0 + 4, 7 - 2) = (4, 5)
F' = F + g = (-2 + 4, 6 - 2) = (2, 4)
G' = G + g = (-3 + 4, 4 - 2) = (1, 2)
H' = H + g = (-2 + 4, 3 - 2) = (2, 1)
3. Plot the new vertices to obtain the image of the pentagon under the translation by the vector g. Join the new vertices in the same order to form the translated pentagon DE'F'G'H'.
The translated pentagon would have vertices:
D'(4, 3), E'(4, 5), F'(2, 4), G'(1, 2), H'(2, 1).
When you plot these points and join them, you will get the image of the original pentagon after the translation by the vector g = <4, -2>.