Question

the graph of a logarithmic function is given. select the function for each graph from the following options. drag each function given above into the area below the appropriate graph, depending on which function is represented by which graph.
functions:
h(x) = 1 - \log_{7} x
f(x) = -\log_{7} x
f(x) = \log_{7} x
g(x) = \log_{7}(x - 1)
g(x) = \log_{7}(-x)
h(x) = \log_{7} x - 1
graphs (labeled 47, 48, 49, 50, 51, 52) with coordinate grids and zoom buttons.

Explanation:

Step1: Analyze the parent function \( f(x)=\log_7 x \)

The parent logarithmic function \( y = \log_b x \) (here \( b = 7>1 \)) has a vertical asymptote at \( x = 0 \), passes through \( (1,0) \), and is increasing (since \( b>1 \)).

Step2: Analyze each transformation:

• Function \( f(x)=\log_7 x \): This is the parent function. Its graph should have a vertical asymptote \( x = 0 \), pass through \( (1,0) \), and be increasing.
• Function \( F(x)=-\log_7 x \): The negative sign reflects the parent function over the \( x \)-axis. So it will be decreasing, with vertical asymptote \( x = 0 \), and pass through \( (1,0) \) (since \( -\log_7 1=0 \)).
• Function \( G(x)=\log_7 (-x) \): The argument \( -x \) means the graph is a reflection of the parent function over the \( y \)-axis. So the domain is \( x<0 \), vertical asymptote \( x = 0 \), and it will be decreasing (since reflecting \( y=\log_7 x \) (increasing) over \( y \)-axis reverses the slope).
• Function \( g(x)=\log_7 (x - 1) \): This is a horizontal shift of the parent function 1 unit to the right. So the vertical asymptote is \( x = 1 \), and it passes through \( (2,0) \) (since \( \log_7(2 - 1)=\log_7 1 = 0 \)), and is increasing.
• Function \( h(x)=\log_7 x-1 \): This is a vertical shift of the parent function 1 unit down. Vertical asymptote \( x = 0 \), passes through \( (1,- 1) \) (since \( \log_7 1-1=0 - 1=-1 \)), and is increasing.
• Function \( H(x)=1-\log_7 x=-\log_7 x + 1 \): Reflect the parent function over the \( x \)-axis (due to \( -\log_7 x \)) and then shift up 1 unit. So it is decreasing, vertical asymptote \( x = 0 \), passes through \( (1,1) \) (since \( 1-\log_7 1 = 1-0 = 1 \)).

Step3: Match with the graphs (assuming the graphs are labeled 47 - 52, we analyze typical features):

• For the graph of \( f(x)=\log_7 x \) (parent, increasing, \( x=0 \) asymptote, passes through \( (1,0) \)): Look for an increasing graph with vertical asymptote \( x = 0 \) and passing through \( (1,0) \).
• For \( F(x)=-\log_7 x \) (decreasing, \( x = 0 \) asymptote, passes through \( (1,0) \)): Decreasing graph, \( x = 0 \) asymptote, \( (1,0) \).
• For \( G(x)=\log_7 (-x) \) (decreasing, \( x = 0 \) asymptote, domain \( x<0 \)): Graph on the left side of \( y \)-axis ( \( x<0 \) ), decreasing.
• For \( g(x)=\log_7 (x - 1) \) (increasing, vertical asymptote \( x = 1 \), passes through \( (2,0) \)): Asymptote at \( x = 1 \), increasing, passes through \( (2,0) \).
• For \( h(x)=\log_7 x-1 \) (increasing, vertical asymptote \( x = 0 \), passes through \( (1,-1) \)): Increasing, \( x = 0 \) asymptote, \( (1,-1) \).
• For \( H(x)=1-\log_7 x \) (decreasing, vertical asymptote \( x = 0 \), passes through \( (1,1) \)): Decreasing, \( x = 0 \) asymptote, \( (1,1) \).

(Note: Since the exact graph details (like grid points) are a bit unclear from the image, but using transformation rules, we can match each function to its graph. For example, the graph with asymptote \( x = 1 \) must be \( g(x)=\log_7(x - 1) \), the one with reflection over \( x \)-axis (decreasing through \( (1,0) \)) is \( F(x)=-\log_7 x \), etc.)

Answer:

To fully answer, we need to match each graph (47 - 52) with the functions:

• Graph 47 (assuming it's the increasing one through (1,0)): \( f(x)=\log_7 x \)
• Graph 48 (assuming it's increasing with asymptote \( x = 1 \)): \( g(x)=\log_7(x - 1) \)
• Graph 49 (assuming it's decreasing through (1,0)): \( F(x)=-\log_7 x \)
• Graph 50 (assuming it's increasing through (1, - 1)): \( h(x)=\log_7 x-1 \)
• Graph 51 (assuming it's decreasing through (1,1)): \( H(x)=1-\log_7 x \)
• Graph 52 (assuming it's on the left of \( y \)-axis, decreasing): \( G(x)=\log_7(-x) \)

(The exact matching depends on the visual details of the graphs, but the above is based on transformation rules for logarithmic functions.)