Question

Graph a transformation for the function:Linear : y=xExpress the equation using function notation and x-y notation.

Answer 1

If a linear equation y = x is translated h unit to the right and k units up, then

The equation will be

`\begin{gathered} y=(x-h)+k \\ f(x)=(x-h)+k \end{gathered}`

If a linear equation y = x is translated h unit to the left and k units down, then

The equation will be

`\begin{gathered} y=(x+h)-k \\ f(x)=(x+h)-k \end{gathered}`

If a linear equation y = x is translated h unit to the right and k units down, then

The equation will b

`\begin{gathered} y=(x-h)-k \\ f(x)=(x-h)-k \end{gathered}`

If a linear equation y = x is translated h unit to the left and k units up, then

The equation will be

`\begin{gathered} y=(x+h)+k \\ f(x)=(x+h)+k \end{gathered}`

Example:

The parent function is y = x is translated 2 units right and 1 unit down, then

It will be

`\begin{gathered} h=2,k=-1 \\ y=(x-2)-1 \end{gathered}`

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

`y=(1-2)-1=-2`

The first point is (1, -2)

Put x = -1

`y=(-1-2)-1=-4`

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

`y=(-x)`

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