Question

left and right, up and down
lets explore how you know whether a translation moves a figure to the left or to the right or up or down.
1 suppose you translate the point (x, y) horizontally c units.
(a) write the coordinates of the image of the point.
(b) how do you know whether the translation of the point is to the left or to the right?
2 suppose you translate the point (x, y) vertically d units.
(a) write the coordinates of the image of the point.
(b) how do you know whether the translation of the point is up or down?
3 suppose you repeatedly translate a point up 1 unit and right 2 units. does the point remain on a straight line as you translate it? draw an example to explain your answer.

Explanation:

Step1: Horizontal translation

When a point $(x,y)$ is translated horizontally $c$ units, the $y -$coordinate remains the same and the $x -$coordinate changes. The new coordinates are $(x + c,y)$.

Step2: Direction of horizontal translation

If $c>0$, the point moves to the right because we are adding a positive value to the $x -$coordinate. If $c < 0$, the point moves to the left since we are adding a negative value to the $x -$coordinate.

Step3: Vertical translation

When a point $(x,y)$ is translated vertically $d$ units, the $x -$coordinate remains the same and the $y -$coordinate changes. The new coordinates are $(x,y + d)$.

Step4: Direction of vertical translation

If $d>0$, the point moves up because we are adding a positive value to the $y -$coordinate. If $d < 0$, the point moves down since we are adding a negative value to the $y -$coordinate.

Step5: Repeated translation analysis

Let the initial point be $(x_0,y_0)$. After the first translation, the point becomes $(x_0 + 2,y_0+1)$. After the second translation, it becomes $(x_0 + 2\times2,y_0 + 1\times2)$ and so on. The slope between any two of the translated - points is $m=\frac{\Delta y}{\Delta x}=\frac{1}{2}$. Since the slope between any two of the translated points is constant, the points lie on a straight line.

Answer:

1.
a. $(x + c,y)$
b. If $c>0$, move right; if $c < 0$, move left.
2.
a. $(x,y + d)$
b. If $d>0$, move up; if $d < 0$, move down.

1. Yes, the point remains on a straight - line. The slope between any two of the translated points is $\frac{1}{2}$. For example, if the initial point is $(0,0)$, after the first translation it is $(2,1)$, after the second it is $(4,2)$. The line passing through $(0,0)$ and $(2,1)$ has the equation $y=\frac{1}{2}x$, and all the translated points will lie on this line.