write equation of given translation of y=|x|
1 unit down 5 units right
10 units left
4 units up 6 units left
To write the equations for the given translations of y = |x|, we need to apply each translation step-by-step.
Translation: 1 unit down, 5 units right
To translate 1 unit down, we subtract 1 from the original equation. To translate 5 units right, we subtract 5 from the x-coordinate.
The equation for this translation is:
y = |x - 5| - 1
Translation: 10 units left
To translate 10 units left, we add 10 to the x-coordinate.
The equation for this translation is:
y = |x - 5 + 10| - 1
Simplifying it:
y = |x + 5| - 1
Translation: 4 units up, 6 units left
To translate 4 units up, we add 4 to the y-coordinate. To translate 6 units left, we add 6 to the x-coordinate.
The equation for this translation is:
y = |x + 5 + 6| - 1 + 4
Simplifying it:
y = |x + 11| + 3
Therefore, the equation for the given translations of y = |x| is:
y = |x + 11| + 3
To find the equation of the translated function, we need to understand how translations affect the original function.
The equation y = |x| represents the absolute value function, which is a V-shaped graph centered at the origin.
Translate 1 unit down and 5 units right:
To shift the graph 1 unit down, subtract 1 from the original function y = |x|.
To shift the graph 5 units right, subtract 5 from x.
Therefore, the translated function is y = |x - 5| - 1.
Translate 10 units left:
To shift the graph 10 units left, add 10 to x.
Therefore, the translated function after the second translation is y = |x - 5 + 10| - 1, which simplifies to y = |x + 5| - 1.
Translate 4 units up and 6 units left:
To shift the graph 4 units up, add 4 to the translated function y = |x + 5| - 1.
To shift the graph 6 units left, add 6 to x.
Therefore, the final translated function after all the given translations is y = |x + 5 - 6| - 1 + 4, which simplifies to y = |x - 1| + 3.