Question

given the function $f(x) = \log_7 x$, which of the following functions is the transformation of $f(x)$ 9 units down, 5 units left, reflected across the x - axis, and stretched vertically by a factor of 8? (1 point)\
\\(\circ\\) $g(x) = -8\log_7(x + 5) - 9$\
\\(\circ\\) $g(x) = 8\log_7(-x + 5) - 9$\
\\(\circ\\) $g(x) = -8\log_7(x - 5) - 9$\
\\(\circ\\) $g(x) = -8\log_7(x + 9) - 5$

Explanation:

Step1: Recall transformation rules

For a logarithmic function \( y = \log_b(x) \), the transformations are as follows:

• Vertical stretch by a factor of \( a \): \( y = a\log_b(x) \)
• Reflection across the x - axis: \( y=-a\log_b(x) \)
• Horizontal shift \( h \) units (left if \( h>0 \), right if \( h < 0 \)): \( y=-a\log_b(x + h) \)
• Vertical shift \( k \) units (up if \( k>0 \), down if \( k < 0 \)): \( y=-a\log_b(x + h)+k \)

Step2: Apply each transformation

• Vertical stretch by a factor of 8: Start with \( f(x)=\log_7x \), after vertical stretch, we get \( y = 8\log_7x \)
• Reflection across the x - axis: After reflection, the function becomes \( y=-8\log_7x \)
• Horizontal shift 5 units left: For a horizontal shift \( h = 5 \) units left, we replace \( x \) with \( x + 5 \) in the function. So the function becomes \( y=-8\log_7(x + 5) \)
• Vertical shift 9 units down: For a vertical shift \( k=-9 \) (down 9 units), we add - 9 to the function. So the function becomes \( y=-8\log_7(x + 5)-9 \)

Answer:

\( g(x)=-8\log_7(x + 5)-9 \) (the first option)