Absolute Value Function
To graph an absolute value function such as f(x) = x + 4 we could
calculate several ordered pairs, plot them, and then connect the points.
However, the graph of f(x) = x + 4 is related to the graph of f(x) = x.
It has the same characteristic
shape, but the
+4 outside the absolute
value symbols shifts the graph up 4 units.
Likewise, the graph of f(x) = x + 3 is the same as the graph of
f(x) = x but shifted down 3 units.
Example
Determine the equation of Line A and Line B shown in the graph.
State the domain and range of each function.
Solution
Line A is the graph of f(x) = x shifted up 2 units. Therefore, its equation
is f(x) = x + 2. The domain is all real numbers; the range is y
≥ 2.
Line B is the graph of f(x) = x shifted down 4 units. Therefore, its
equation is f(x) = x  4. The domain is all real numbers; the range is
y ≥ 4.
When a constant is added or subtracted inside the absolute value symbols
the graph of f(x) = x is also shifted, but this time to the left or right.
For example, the graph of f(x) = x  3 has the same shape as that of
f(x) = x but it is shifted 3 units to the right.
Likewise, the graph of f(x) = x + 4 has the same shape as f(x) = x but it
is shifted 4 units to the left.
Be careful.
The graph of y = x  3 is
shifted 3 units to the right; that is, in the
positive direction along the xaxis. This
may be the opposite of what you expect.
