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what kind of transformation converts the graph of f(x) = -|x|+2 into the graph of g(x)=-4|x|+8
Answer. ThIS. Question.
what kind of transformation converts the graph of f(x) = -|x|+2 into the graph of g(x)=-4|x|+8
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GPT-4o mini
To convert the graph of \( f(x) = -|x| + 2 \) into the graph of \( g(x) = -4|x| + 8 \), we can analyze the transformations step by step:
1. **Vertical Stretch**: The coefficient of the absolute value function in \( g(x) \) is -4 instead of -1 in \( f(x) \). This means that the graph of \( f(x) \) is stretched vertically by a factor of 4 and reflected (since the coefficient is negative). So, we take \( f(x) \) and stretch it vertically:
\[ f'(x) = -4|x| + c \]
where \( c \) is some constant that we need to adjust.
2. **Vertical Translation**: After stretching, we need to move (translate) the graph upwards to match the constant term in \( g(x) \). The original graph \( f(x) \) has a maximum value of 2 (when \( x = 0 \)), and we want the new maximum (from the transformation) to be 8.
Since we have stretched \( -|x| \) to \( -4|x| \), the maximum value after the stretch is now \( 0 \) (at \( x = 0 \)). To achieve the value of 8, we increase the entire graph by 8 units.
Combining these transformations, we can summarize that:
- **Step 1**: Vertically stretch by a factor of 4 (making it \( -4|x| \)).
- **Step 2**: Vertically translate up by 8 units (resulting in \( -4|x| + 8 \)).
In conclusion, the transformation involves a vertical stretch by a factor of 4 followed by a vertical shift upwards by 6 units.
1. **Vertical Stretch**: The coefficient of the absolute value function in \( g(x) \) is -4 instead of -1 in \( f(x) \). This means that the graph of \( f(x) \) is stretched vertically by a factor of 4 and reflected (since the coefficient is negative). So, we take \( f(x) \) and stretch it vertically:
\[ f'(x) = -4|x| + c \]
where \( c \) is some constant that we need to adjust.
2. **Vertical Translation**: After stretching, we need to move (translate) the graph upwards to match the constant term in \( g(x) \). The original graph \( f(x) \) has a maximum value of 2 (when \( x = 0 \)), and we want the new maximum (from the transformation) to be 8.
Since we have stretched \( -|x| \) to \( -4|x| \), the maximum value after the stretch is now \( 0 \) (at \( x = 0 \)). To achieve the value of 8, we increase the entire graph by 8 units.
Combining these transformations, we can summarize that:
- **Step 1**: Vertically stretch by a factor of 4 (making it \( -4|x| \)).
- **Step 2**: Vertically translate up by 8 units (resulting in \( -4|x| + 8 \)).
In conclusion, the transformation involves a vertical stretch by a factor of 4 followed by a vertical shift upwards by 6 units.