To graph the function \( f(x) = -8(6x - 13) - 16 \), we can identify the transformations applied to the basic function. Let's break it down step-by-step:
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Start with the basic function: The innermost part is \( 6x \). This represents a linear function.
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Horizontal scaling: The \( 6x \) indicates a horizontal compression by a factor of \( 1/6 \).
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Horizontal translation: The constant \( -13 \) in \( 6x - 13 \) indicates a horizontal shift to the right by \( 13/6 \) units.
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Vertical scaling and reflection: The coefficient of \( -8 \) means we are vertically stretching the graph of the function by a factor of \( 8 \) and reflecting it across the x-axis (because of the negative sign).
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Vertical translation: The \( -16 \) at the end shifts the entire graph downward by \( 16 \) units.
So, the transformations can be summarized as follows:
- Horizontal compression by a factor of \( \frac{1}{6} \)
- Horizontal shift to the right by \( \frac{13}{6} \)
- Reflection across the x-axis (due to the negative coefficient)
- Vertical stretch by a factor of \( 8 \)
- Vertical shift downward by \( 16 \) units
By applying these transformations step-by-step to the parent function \( f(x) = x \), you can construct the graph of the function \( f(x) = -8(6x - 13) - 16 \).