First, you have to find the translation made to (4,-9) to determine the point (9,-14). You can plot both points to see their positions in the coordinate system
As you can see, the image (9,-14) is to the right and downwards of the preimage (4,-9), this means that the point was moved to the right and down.
- To make a horizontal movement to the right, you have to add a constant "c" to the x-coordinate of the point, following the rule:
[tex](x,y)\to(x+c,y)[/tex]To determine the value of the constant, you have to compare the x-coordinates of both points:
[tex]4+c=9[/tex]Subtract 4 to both sides of the expression
[tex]\begin{gathered} 4-4+c=9-4 \\ c=5 \end{gathered}[/tex]The point was moved 5 units to the right.
- To make a vertical translation down, you have to subtract a constant "k" to the y-coordinate of the point, following the rule
[tex](x,y)\to(x,y-k)[/tex]To determine the number of units of the vertical translation, you have to compare the y-coordinates of both points:
[tex]-9-k=-14[/tex]Add 9 to both sides of the expression
[tex]\begin{gathered} -9+9-k=-14+9 \\ -k=-5 \end{gathered}[/tex]Multiply both sides by -1 to change the sign:
[tex]\begin{gathered} (-1)(-k)=(-1)(-5) \\ k=5 \end{gathered}[/tex]The point was moved 5 units down.
Next, you have to apply the same translation to point (-9,-8), as mentioned, the point was moved 5 units to the right and 5 units down, the translation rule is:
[tex](x,y)\to(x+5,y-5)[/tex]Add 5 to the x-coordinate and subtract 5 to the y-coordinate:
[tex](-9,-8)\to(-9+5,-8-5)=(-4,-13)[/tex]The image of the point (-9,-8) after the transformation will be (-4,-13)
