Question: Use a table to perform the reflection of $y = f(x)$ across the y-axis. Graph the function and its reflection.
Graph shape: a smooth curve on the left side of the y-axis, starting near $(-5, 3)$, dipping to a minimum around $(-3.8, -1.5)$, then rising to a maximum around $(-1.2, 2.7)$, and ending near $(0, 1)$.
1. **State the problem:** We need to reflect the function $y = f(x)$ across the y-axis and graph both the original function and its reflection.
2. **Reflection rule:** Reflecting a function across the y-axis means replacing $x$ by $-x$. The reflected function is $y = f(-x)$.
3. **Original points:** Given points on $y = f(x)$ are approximately:
| $x$ | $y = f(x)$ |
|-----|------------|
| -5 | 3 |
| -3.8| -1.5 |
| -1.2| 2.7 |
| 0 | 1 |
4. **Reflection points:** For each original point $(x, y)$, the reflected point is $(-x, y)$:
| $x$ | $y = f(x)$ | Reflected point $(-x, y)$ |
|------|------------|---------------------------|
| -5 | 3 | $(5, 3)$ |
| -3.8 | -1.5 | $(3.8, -1.5)$ |
| -1.2 | 2.7 | $(1.2, 2.7)$ |
| 0 | 1 | $(0, 1)$ |
5. **Summary:** The reflection flips the curve from the left side of the y-axis to the right side, preserving the $y$ values but changing the sign of $x$.
6. **Graphing:** Plot the original points on the left side and the reflected points on the right side, connecting smoothly to show the curve and its reflection.
**Final answer:** The reflected function is $y = f(-x)$ with points as above, showing a smooth curve mirrored across the y-axis.