If a linear equation y = x is translated h unit to the right and k units up, then
The equation will be
[tex]\begin{gathered} y=(x-h)+k \\ f(x)=(x-h)+k \end{gathered}[/tex]
If a linear equation y = x is translated h unit to the left and k units down, then
The equation will be
[tex]\begin{gathered} y=(x+h)-k \\ f(x)=(x+h)-k \end{gathered}[/tex]
If a linear equation y = x is translated h unit to the right and k units down, then
The equation will b
[tex]\begin{gathered} y=(x-h)-k \\ f(x)=(x-h)-k \end{gathered}[/tex]
If a linear equation y = x is translated h unit to the left and k units up, then
The equation will be
[tex]\begin{gathered} y=(x+h)+k \\ f(x)=(x+h)+k \end{gathered}[/tex]
Example:
The parent function is y = x is translated 2 units right and 1 unit down, then
It will be
[tex]\begin{gathered} h=2,k=-1 \\ y=(x-2)-1 \end{gathered}[/tex]
To graph it, substitute x by any values and calculate the corresponding values of y then draw the points and join them
Put x = 1
[tex]y=(1-2)-1=-2[/tex]
The first point is (1, -2)
Put x = -1
[tex]y=(-1-2)-1=-4[/tex]
The second point is (-1, -4)
Plot them on the graph and join them
This is the graph of both equations
The green is the graph of y = x
The black is the graph of y = (x - 2) - 1
If we reflect y = x on the y-axis we will change the sign of x, then the equation will be
[tex]y=(-x)[/tex]
Then the graph will be
The purple line represents the equation y = x after reflection on the y-axis
y = (-x)
If we want to reflect the equation after translation we just change the sign of x
y = (-x - 2) - 1
The black lines represent them
Look at the point (1, -2) it changes to (-1, -2) and the point (-1, -4) changes to (1, -4)
We just change the sign of x when we reflect on the y-axis
