define plot `y=|x-2|+3|x-3|`

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Answer

To plot the graph of the function \( y = |x - 2| + 3|x - 3| \), we will analyze the function by breaking it down into intervals based on the points where the expressions inside the absolute values change sign.

### Step 1: Identify Critical Points
The critical points occur where the expressions inside the absolute values equal zero:
- \( |x - 2| = 0 \) at \( x = 2 \)
- \( |x - 3| = 0 \) at \( x = 3 \)

These points divide the number line into three intervals:
1. \( x < 2 \)
2. \( 2 \leq x < 3 \)
3. \( x \geq 3 \)

### Step 2: Analyze Each Interval

**Interval 1: \( x < 2 \)**
- Here, both \( x - 2 \) and \( x - 3 \) are negative.
- Thus, \( |x - 2| = -(x - 2) = -x + 2 \) and \( |x - 3| = -(x - 3) = -x + 3 \).
- The function becomes:
\[ y = -x + 2 + 3(-x + 3) = -x + 2 - 3x + 9 = -4x + 11 \]

**Interval 2: \( 2 \leq x < 3 \)**
- Here, \( x - 2 \) is non-negative and \( x - 3 \) is negative.
- Thus, \( |x - 2| = x - 2 \) and \( |x - 3| = -(x - 3) = -x + 3 \).
- The function becomes:
\[ y = (x - 2) + 3(-x + 3) = x - 2 - 3x + 9 = -2x + 7 \]

**Interval 3: \( x \geq 3 \)**
- Here, both \( x - 2 \) and \( x - 3 \) are non-negative.
- Thus, \( |x - 2| = x - 2 \) and \( |x - 3| = x - 3 \).
- The function becomes:
\[ y = (x - 2) + 3(x - 3) = x - 2 + 3x - 9 = 4x - 11 \]

### Step 3: Summary of Piecewise Function
The piecewise function can be summarized as:
\[ y = \begin{cases} -4x + 11 & \text{if } x < 2 \\ -2x + 7 & \text{if } 2 \leq x < 3 \\ 4x - 11 & \text{if } x \geq 3 \end{cases} \]

### Step 4: Find Points for Plotting
- For \( x = 2 \):
\[ y = -2(2) + 7 = 3 \quad \Rightarrow \quad (2, 3) \]
- For \( x = 3 \):
\[ y = 4(3) - 11 = 1 \quad \Rightarrow \quad (3, 1) \]

### Step 5: Plot the Points and Draw the Lines
1. Plot the points \( (2, 3) \) and \( (3, 1) \).
2. Draw the line for \( y = -4x + 11 \) for \( x < 2 \).
3. Draw the line for \( y = -2x + 7 \) for \( 2 \leq x < 3 \).
4. Draw the line for \( y = 4x - 11 \) for \( x \geq 3 \).