Question

which of the following transformations are used when transforming the graph of the parent function $f(x)=log_{5}x$ to the graph of $g(x)=-2log_{5}(x - 6)$? select all that apply.
(1 point)

• shift the graph of $f(x)$ 6 units down.
• shift the graph of $f(x)$ 6 units to the left
• reflect the graph of $f(x)$ over the $x$-axis
• stretch the graph of $f(x)$ vertically by a factor of 2
• stretch the graph of $f(x)$ vertically by a factor of $-\frac{1}{2}$

Explanation:

Step1: Analyze horizontal shift

For a function \( y = f(x - h) \), the graph shifts \( h \) units to the right if \( h>0 \). Here, \( g(x)=\log_5(x - 6) \) part (before vertical transformations) is \( f(x-6) \) where \( f(x)=\log_5x \), so it shifts 6 units to the right (not left or down for horizontal/vertical shift here yet). So "Shift the graph of \( f(x) \) 6 units down" and "Shift the graph of \( f(x) \) 6 units to the left" are wrong.

Step2: Analyze reflection

The negative sign in front of the log function (\( -2\log_5(x - 6) \)) indicates a reflection over the \( x \)-axis. So "Reflect the graph of \( f(x) \) over the \( x \)-axis" is correct.

Step3: Analyze vertical stretch

The coefficient 2 in \( -2\log_5(x - 6) \) (ignoring sign for stretch factor) means a vertical stretch by a factor of 2. The negative sign is for reflection. So "Stretch the graph of \( f(x) \) vertically by a factor of 2" is correct. "Stretch the graph of \( f(x) \) vertically by a factor of \( -\frac{1}{2} \)" is wrong as stretch factor is positive magnitude here.

Answer:

• Reflect the graph of \( f(x) \) over the \( x \)-axis
• Stretch the graph of \( f(x) \) vertically by a factor of 2