To analyze the reflection of the function \( f(x) = 2(x + 22) \), we need to first understand the transformation that occurs when we define \( g(x) = -f(-x) \).
Step 1: Calculate \( f(x) \)
We begin by finding the values of \( f(x) \) for some values of \( x \):
-
\( x = -22 \): \[ f(-22) = 2(-22 + 22) = 2(0) = 0 \]
-
\( x = -20 \): \[ f(-20) = 2(-20 + 22) = 2(2) = 4 \]
-
\( x = -10 \): \[ f(-10) = 2(-10 + 22) = 2(12) = 24 \]
-
\( x = 0 \): \[ f(0) = 2(0 + 22) = 2(22) = 44 \]
-
\( x = 10 \): \[ f(10) = 2(10 + 22) = 2(32) = 64 \]
So, we can summarize the values of \( f(x) \):
- \( f(-22) = 0 \)
- \( f(-20) = 4 \)
- \( f(-10) = 24 \)
- \( f(0) = 44 \)
- \( f(10) = 64 \)
Step 2: Calculate \( g(x) = -f(-x) \)
Now, let's calculate \( g(x) \) based on \( f(-x) \):
- \( g(-22) = -f(22) = -2(22 + 22) = -2(44) = -88 \)
- \( g(-20) = -f(20) = -2(20 + 22) = -2(42) = -84 \)
- \( g(-10) = -f(10) = -2(10 + 22) = -2(32) = -64 \)
- \( g(0) = -f(0) = -2(0 + 22) = -2(22) = -44 \)
- \( g(10) = -f(-10) = -2(-10 + 22) = -2(12) = -24 \)
Step 3: Summarize \( g(x) \)
The results for \( g(x) \) are:
- \( g(-22) = -88 \)
- \( g(-20) = -84 \)
- \( g(-10) = -64 \)
- \( g(0) = -44 \)
- \( g(10) = -24 \)
Step 4: Resulting Table for \( g(x) \)
The resulting coordinates for \( g(x) \) compared to \( f(x) \): | \( x \) | \( f(x) \) | \( g(x) \) | |----------|-------------|---------------| | -22 | 0 | -88 | | -20 | 4 | -84 | | -10 | 24 | -64 | | 0 | 44 | -44 | | 10 | 64 | -24 |
Conclusion
The table shown above reflects the transformation that \( g(x) = -f(-x) \) represents, confirming that it results in a reflection about the origin of the graphed function \( f(x) \). Thus, the \( g(x) \) values correctly show that the graph of \( g(x) \) reflects \( f(x) \) through the origin.
You can plot these \( g(x) \) points to visualize the reflection.
1 answer