Linear translations

Answer 1

Is your question asking to explain linear translation? If so, we need to understand what linear transformation is first. Linear transformation is essentially changing an object's shape through translation, reflection, and stretching on an initial function.

Visualize the initial function f(x)=x an example of a transformation would be f(x)=x transforming to f(x)=x+1

To rename the function after this transformation we would use g(x)=f(x)+1

From this we can easily see that the the function f(x) moves up one unit.

Now back to translation! During translation a function is able to move vertically along the y-axis or horizontally along the x-axis. Here are a few rules to remember:

1. Horizontal Translation: Description goes inside function ex: f(x+5)
2. Shift left (+) Shift right (-) ex: f(x+5) is shifted 5 units to the left
3. Vertical Translation: Description goes outside of the function ex: f(x)+5
4. Shift up (+) Shift down (-)

Imagine the initial function f(x)=x again. Let's say that initial function moves horizontally 7 units to the right, we would describe the function as f(x-7). Now lets say the initial function gets moved up 3 units, we would describe the function as f(x)+3

Now Imagine the function f(x)=x-2 shifts 3 units left. We can solve for g(x) easily as follows

g(x)=f(x+3)

g(x)=x+3-2

g(x)=x+1

Now Imagine the function f(x)=x-2 shifts 3 units down. We can solve for g(x) easily as follows

g(x)=f(x)-3

g(x)=x-2-3

g(x)=x-5

Hope this helps!