Understanding the Function f(x) = |x – 2| + 1: Step-by-Step Explanation and Graph Analysis

f(x) = |x – 2| + 1

To understand the function f(x) = |x – 2| + 1, let’s break it down step-by-step

To understand the function f(x) = |x – 2| + 1, let’s break it down step-by-step.

The function f(x) involves three main components: the absolute value function, the expression (x – 2), and adding 1 to the result.

1. Absolute Value Function:
The absolute value of a number is its distance from zero on a number line. It is denoted by two vertical lines around the number. For example, |3| = 3 and |-4| = 4. The absolute value function ensures that the resulting value is always non-negative.

2. Expression (x – 2):
Here, we have the expression (x – 2). This means we are subtracting 2 from the value of x. The expression inside the absolute value function tells us how far x is from 2.

3. Adding 1:
After taking the absolute value and subtracting 2 from x, we add 1 to the final result.

Now, let’s understand how this all comes together by analyzing a few values.

1. x = 0:
When x = 0, the expression (x – 2) becomes (0 – 2) = -2. We then take the absolute value: |(-2)| = 2. Finally, we add 1 to get the value of f(x) = 2 + 1 = 3.

2. x = 3:
When x = 3, the expression (x – 2) becomes (3 – 2) = 1. We take the absolute value: |1| = 1. Finally, we add 1 to get the value of f(x) = 1 + 1 = 2.

3. x = 2:
When x = 2, the expression (x – 2) becomes (2 – 2) = 0. Since the absolute value of 0 is 0, we have |0| = 0. Finally, we add 1 to get the value of f(x) = 0 + 1 = 1.

4. x = 5:
When x = 5, the expression (x – 2) becomes (5 – 2) = 3. We take the absolute value: |3| = 3. Finally, we add 1 to get the value of f(x) = 3 + 1 = 4.

By analyzing these examples and any other values, we can see that the graph of f(x) = |x – 2| + 1 represents a V-shape or an upside-down V-shape depending on the values of x. The vertex of this V-shaped graph is located at x = 2.

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