Draw the labled graph of following
(i) `y = |7-2x|` , (ii) `y = |x-1|-|3x-2|` , (iii) `y= |x-1|+|x-4| + |x-7|`

Answer

To solve the problem of drawing the labeled graphs for the given functions, we will break down each function step by step.

### (i) Graph of `y = |7 - 2x|`

1. **Identify the critical point**: The expression inside the absolute value changes sign when \( 7 - 2x = 0 \). Solving for \( x \):
\[ 7 - 2x = 0 \implies 2x = 7 \implies x = \frac{7}{2} = 3.5 \]

2. **Define the piecewise function**:
- For \( x < 3.5 \):
\[ y = 7 - 2x \]
- For \( x \geq 3.5 \):
\[ y = 2x - 7 \]

3. **Find the y-intercept and x-intercept**:
- **y-intercept**: Set \( x = 0 \):
\[ y = 7 - 2(0) = 7 \quad \text{(point: (0, 7))} \]
- **x-intercept**: Set \( y = 0 \):
\[ 0 = 7 - 2x \implies 2x = 7 \implies x = 3.5 \quad \text{(point: (3.5, 0))} \]

4. **Plot the points and draw the lines**:
- Plot the points (0, 7) and (3.5, 0).
- Draw the line for \( y = 7 - 2x \) from \( x = 0 \) to \( x = 3.5 \).
- Draw the line for \( y = 2x - 7 \) starting from \( x = 3.5 \) onward.

### (ii) Graph of `y = |x - 1| - |3x - 2|`

1. **Identify critical points**: The expressions change sign at \( x = 1 \) and \( x = \frac{2}{3} \).

2. **Define the piecewise function**:
- For \( x < \frac{2}{3} \):
\[ y = -(x - 1) - (-(3x - 2)) = -x + 1 + 3x - 2 = 2x - 1 \]
- For \( \frac{2}{3} \leq x < 1 \):
\[ y = -(x - 1) - (3x - 2) = -x + 1 - 3x + 2 = -4x + 3 \]
- For \( x \geq 1 \):
\[ y = (x - 1) - (3x - 2) = x - 1 - 3x + 2 = -2x + 1 \]

3. **Find intercepts**:
- **y-intercept**: Set \( x = 0 \):
\[ y = |0 - 1| - |3(0) - 2| = 1 - 2 = -1 \quad \text{(point: (0, -1))} \]
- **x-intercepts**: Set \( y = 0 \) for each piece:
- For \( 2x - 1 = 0 \): \( x = \frac{1}{2} \)
- For \( -4x + 3 = 0 \): \( x = \frac{3}{4} \)
- For \( -2x + 1 = 0 \): \( x = \frac{1}{2} \)

4. **Plot the points and draw the lines**:
- Plot the points (0, -1), (0.5, 0), (0.75, 0), and (1, -1).
- Draw the lines for each piecewise function.

### (iii) Graph of `y = |x - 1| + |x - 4| + |x - 7|`

1. **Identify critical points**: The expressions change sign at \( x = 1, 4, 7 \).

2. **Define the piecewise function**:
- For \( x < 1 \):
\[ y = -(x - 1) - (x - 4) - (x - 7) = -3x + 12 \]
- For \( 1 \leq x < 4 \):
\[ y = (x - 1) - (x - 4) - (x - 7) = -x + 10 \]
- For \( 4 \leq x < 7 \):
\[ y = (x - 1) + (x - 4) - (x - 7) = x + 2 \]
- For \( x \geq 7 \):
\[ y = (x - 1) + (x - 4) + (x - 7) = 3x - 12 \]

3. **Find intercepts**:
- **y-intercept**: Set \( x = 0 \):
\[ y = |0 - 1| + |0 - 4| + |0 - 7| = 1 + 4 + 7 = 12 \quad \text{(point: (0, 12))} \]
- **x-intercepts**: Set \( y = 0 \) for each piece:
- For \( -3x + 12 = 0 \): \( x = 4 \)
- For \( -x + 10 = 0 \): \( x = 10 \)
- For \( x + 2 = 0 \): \( x = -2 \)
- For \( 3x - 12 = 0 \): \( x = 4 \)

4. **Plot the points and draw the lines**:
- Plot the points (0, 12), (1, 0), (4, 0), and (7, 0).
- Draw the lines for each piecewise function.