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 1 unit 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 x - coordinates and y - coordinates of corresponding points of Figure 1 and Figure 2, we can see that the x - coordinate of each point in Figure 1 decreases by 7 (e.g., if a point in Figure 1 has x - coordinate \(x_1\) and the corresponding point in Figure 2 has \(x_2\), \(x_2=x_1 - 7\)) and the y - coordinate decreases by 2 (e.g., \(y_2=y_1 - 2\)). So the translation is 7 units left and 2 units down.

Step2: Translate Figure 1 for part (b)

Let the original coordinates of a point in Figure 1 be \((x,y)\). When translated 1 unit horizontally (assuming to the right, so \(x'=x + 1\)) and - 8 units vertically (downward, so \(y'=y-8\)). We need to apply this transformation to each vertex of Figure 1. For example, if a vertex of Figure 1 has coordinates \((2,8)\), after translation, the new coordinates are \((2 + 1,8-8)=(3,0)\).

Step3: Explain coordinate - determination in part (b)

We use the rules of horizontal and vertical translations. For a horizontal translation by \(h\) units, the x - coordinate of a point \((x,y)\) changes to \(x+h\) (right if \(h>0\), left if \(h < 0\)), and for a vertical translation by \(k\) units, the y - coordinate of a point \((x,y)\) changes to \(y + k\) (up if \(k>0\), down if \(k < 0\)). Here \(h = 1\) and \(k=-8\).

Step4: Graphing for part (d)

Plot the new - coordinates of the vertices of Figure 1 obtained in part (b) on the coordinate plane and connect them to form Figure 3. This visually verifies the translation.

Answer:

a. 7 units left and 2 units down.
b. Apply \((x,y)\to(x + 1,y-8)\) to each vertex of Figure 1 to get new coordinates.
c. Used rules of horizontal (\(x\to x + h\)) and vertical (\(y\to y + k\)) translations with \(h = 1\) and \(k=-8\).
d. Plot new - coordinates of vertices from part (b) to form Figure 3 for verification.