Question

in your own words, explain how horizontal and vertical translations each affect the coordinates of the points of a figure. remember a point with the coordinates (x, y), when translated horizontally by c units, has new coordinates (x + c, y). a point with the coordinates (x, y), when translated vertically by c units, has new coordinates (x, y + c). practice 1 use the figures shown to complete parts (a) through (d). a) describe the sequence of translations used to move figure 1 onto figure 2. b) determine the coordinates of the image of figure 1 when you translate it horizontally and -8 units vertically. c) explain how you determined the coordinates in part (b). d) verify your answer to part (b) by graphing the image. label it figure 3.

Explanation:

Step1: Analyze translation from Figure 1 to Figure 2

By observing the graph, we can see that to move Figure 1 onto Figure 2, we move 6 units to the left (since \(x\) - coordinates change from positive values in Figure 1 to negative values in Figure 2) and 2 units down (since \(y\) - coordinates decrease).

Step2: Translate Figure 1 horizontally and vertically for part (b)

Let the original coordinates of a point in Figure 1 be \((x,y)\). When translated 6 units horizontally (assuming left - hand side translation) and 8 units vertically down, the new coordinates \((x',y')\) are given by \(x'=x - 6\) and \(y'=y-8\). We would need to know the original coordinates of the vertices of Figure 1. Suppose the vertices of Figure 1 are \((x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)\). The new vertices after translation are \((x_1 - 6,y_1 - 8),(x_2 - 6,y_2 - 8),(x_3 - 6,y_3 - 8),(x_4 - 6,y_4 - 8)\).

Step3: Explain coordinate - determination for part (c)

We use the rules of translation. A horizontal translation by \(h\) units changes the \(x\) - coordinate of a point \((x,y)\) to \(x + h\) (if \(h>0\), it's a right - hand side translation; if \(h < 0\), it's a left - hand side translation). A vertical translation by \(k\) units changes the \(y\) - coordinate of a point \((x,y)\) to \(y + k\) (if \(k>0\), it's an upward translation; if \(k < 0\), it's a downward translation). Here \(h=-6\) (6 units to the left) and \(k = - 8\) (8 units down).

Step4: Verify by graphing for part (d)

Plot the original vertices of Figure 1. Then, using the new coordinates calculated in part (b), plot the new points. Connect the new points in the same order as the original figure to form Figure 3 and check if it matches the expected translated figure.

Answer:

a. 6 units left and 2 units down.
b. If the original vertices of Figure 1 are \((x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)\), the new vertices are \((x_1 - 6,y_1 - 8),(x_2 - 6,y_2 - 8),(x_3 - 6,y_3 - 8),(x_4 - 6,y_4 - 8)\)
c. Used the rules of horizontal (\(x\to x+h\)) and vertical (\(y\to y + k\)) translations with \(h=-6\) and \(k=-8\).
d. Plot original and new points as described above to verify.