To sketch the graph of $y = f(x)$ for the interval $[-3, 4]$, let's break it down step by step, keeping in mind that $f(x)$ is a periodic function with a period of 3 and its definition over the interval $-1 < x \leq 2$ is:

$f(x) = |2x - 1| + x$

Step 1: Analyze $f(x)$ over $-1 < x \leq 2$

Let's first compute the values of $f(x) = |2x - 1| + x$ for key points in the interval $-1 < x \leq 2$.

1.1: Break into two cases for $|2x - 1|$

• For $x \leq \frac{1}{2}$, $2x - 1 \leq 0$, so $|2x - 1| = -(2x - 1) = 1 - 2x$.
• For $x > \frac{1}{2}$, $2x - 1 > 0$, so $|2x - 1| = 2x - 1$.

Thus, we have two pieces:

1. $f(x) = (1 - 2x) + x = 1 - x$ for $-1 < x \leq \frac{1}{2}$.
2. $f(x) = (2x - 1) + x = 3x - 1$ for $\frac{1}{2} < x \leq 2$.

Now let's find the values of $f(x)$ at key points:

• At $x = -1$, the interval is open, so no value is defined here.
• At $x = 0$, $f(0) = 1 - 0 = 1$.
• At $x = \frac{1}{2}$, $f\left( \frac{1}{2} \right) = 1 - \frac{1}{2} = \frac{1}{2}$.
• At $x = 1$, $f(1) = 3(1) - 1 = 2$.
• At $x = 2$, $f(2) = 3(2) - 1 = 5$.

Step 2: Extend the function periodically

Since $f(x)$ is periodic with a period of 3, this means we will repeat the pattern for intervals to the left and right of $-1 < x \leq 2$. Specifically:

• For $-4 < x \leq -1$, $f(x) = f(x + 3)$.
• For $2 < x \leq 5$, $f(x) = f(x - 3)$.

Step 3: Calculate values for the extended intervals

3.1: For $-4 < x \leq -1$

Since $f(x)$ repeats every 3 units, calculate $f(x) = f(x + 3)$:

• For $x = -3$, $f(-3) = f(0) = 1$.
• For $x = -2$, $f(-2) = f(1) = 2$.
• For $x = -1$, not defined, since the interval is open at $-1$.

3.2: For $2 < x \leq 5$

Similarly, for $2 < x \leq 5$, we use $f(x) = f(x - 3)$:

• For $x = 3$, $f(3) = f(0) = 1$.
• For $x = 4$, $f(4) = f(1) = 2$.

Step 4: Sketch the graph

1. From $-3 \leq x \leq 0$, the graph repeats the behavior from $-1 < x \leq 2$.
2. From $0 \leq x \leq 3$, use the defined values for $f(x) = |2x - 1| + x$.
3. From $3 \leq x \leq 4$, repeat the earlier values using periodicity.

Final Graph:

The graph will consist of repeating segments of a V-shaped curve with different slopes in each period.

Would you like assistance with a sketching tool or further clarification of the pattern?