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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 x-axis. This may be the opposite of what you expect.

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