Bot GPT 4-o mini To determine which table shows that \( y = kf(x) \) results in the parent function \( y = |x + 4| \) being vertically stretched by a factor of \( k = 3 \), we need to evaluate the function \( f(x) = |x + 4| \) and then apply the stretch.
- Calculate the values of \( f(x) = |x + 4| \):
For \( x = 0 \): \[ f(0) = |0 + 4| = |4| = 4 \]
For \( x = 1 \): \[ f(1) = |1 + 4| = |5| = 5 \]
For \( x = 2 \): \[ f(2) = |2 + 4| = |6| = 6 \]
So, we have:
- When \( x = 0 \), \( f(0) = 4 \)
- When \( x = 1 \), \( f(1) = 5 \)
- When \( x = 2 \), \( f(2) = 6 \)
- Apply the vertical stretch by \( k = 3 \):
Now we find \( y = kf(x) = 3f(x) \).
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For \( x = 0 \): \[ y = 3 \cdot f(0) = 3 \cdot 4 = 12 \]
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For \( x = 1 \): \[ y = 3 \cdot f(1) = 3 \cdot 5 = 15 \]
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For \( x = 2 \): \[ y = 3 \cdot f(2) = 3 \cdot 6 = 18 \]
This gives us:
- \( (0, 12) \)
- \( (1, 15) \)
- \( (2, 18) \)
- Compare with the given tables:
Now we can look at the tables provided to find a match:
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First table:
- (0, 4)
- (1, 5)
- (2, 6)
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Second table:
- (0, -12)
- (1, -15)
- (2, -18)
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Third table:
- (0, 12)
- (1, 15)
- (2, 18) (This matches our calculated values)
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Fourth table:
- (0, 1.3)
- (1, 1.7)
- (2, 2)
The correct table showing that the parent function \( y = |x + 4| \) is vertically stretched by a factor of \( k = 3 \) is the third table:
- \( x \ y = kf(x) \)
- \( 0 \ 12 \)
- \( 1 \ 15 \)
- \( 2 \ 18 \)
1 answer