Answer:
[tex]y = |x - 1| + 3[/tex]
Step-by-step explanation:
The given graph is that of a modulus function. The basic modulus function, y = f(x) = |x|
This will always yield a positive value for f(x).
If x is positive f(x) = x
If x is 0, f(x) = 0
If x is negative, f(x) = -x so f(x) is again positive
The absolute value function |x| is a V-shaped graph that opens upwards and passes through the origin
Horizontal Translation( shift right or left horizontally)
If we translate(shift) this function horizontally to the right by say 1 unit, then the resulting function becomes |x - 1|.
In general, for any function f(x), to translate the graph of f(x) horizontally by k units to the right, we replace x with (x-k) in the equation of the function. The graph of the translated function is given by the equation f(x-k).
[If we replace |x| by |x + k| then the graph is shifted to the left by k units]
Vertical Translation(shift up or down vertically)
If we add a constant m to |x| the graph is translated vertically up or down. Suppose we add the constant 3 to |x|. Then the graph is shifted vertically upward by 3 units and the function becomes |x| + 3
If we subtract 3 units from |x| the graph is shifted downward by 3 units
In general, for any function f(x), to translate the graph of f(x) vertically upward by m units we replace f(x) with f(x) + m
Horizontal and Vertical Translation together
If we shift the graph horizontally right by k units and vertically up by m units we get the resulting function as
|x - k| + m
Looking at the given graph we can see that it is the equivalent of shifting |x| to the right by 1 unit and shifting it upward by 3 units from the original |x| graph
Therefore the given graph is of the form
|x - 1| + 3
The attached image shows the graphs of |x| and |x - 1| + 3 and you can see visually what the explanation is.
