Question

p and building your weekly streak!
use the interactive graph below to sketch a graph of
$y = 4\log_{3}(-x - 3)$.

$\boldsymbol{y}$
9
8
7
6
5
4
3
2
1
$\boldsymbol{x}$
-9 -8 -7 -6 -5 -4 -3 -2
1 2 3 4 5 6 7 8 9
-2
-3
-4
-5
-6
do 4 problems

Explanation:

Answer:

To sketch \( y = 4\log_{3}(-x - 3) \), we analyze transformations of the parent function \( y=\log_{3}(x) \):

Step 1: Analyze the argument of the logarithm

The argument is \( -x - 3=-(x + 3) \). This implies a reflection over the \( y \)-axis (due to the negative sign on \( x \)) and a horizontal shift. The horizontal shift is determined by solving \( -x - 3=0\Rightarrow x=- 3 \), so the vertical asymptote is \( x=-3 \) (compared to \( x = 0 \) for \( y=\log_{3}(x) \)).

Step 2: Analyze the transformations

• Reflection: The negative sign on \( x \) (inside the log) reflects the graph of \( y = \log_{3}(x) \) over the \( y \)-axis.
• Horizontal shift: The \( + 3 \) inside the log (after factoring out the negative sign) shifts the graph 3 units to the left.
• Vertical stretch: The coefficient 4 vertically stretches the graph by a factor of 4.

Step 3: Find key points

For the parent function \( y=\log_{3}(x) \), when \( x = 1 \), \( y = 0 \); when \( x=3 \), \( y = 1 \); when \( x=\frac{1}{3} \), \( y=- 1 \).

We apply the transformations to these points:

1. For the point \( (x,y)=(1,0) \) on \( y = \log_{3}(x) \):

• Reflect over \( y \)-axis: \( (-1,0) \)
• Shift left 3 units: \( (-1-3,0)=(-4,0) \)
• Stretch vertically by 4: \( (-4,0\times4)=(-4,0) \)

1. For the point \( (x,y)=(3,1) \) on \( y=\log_{3}(x) \):

• Reflect over \( y \)-axis: \( (-3,1) \)
• Shift left 3 units: \( (-3 - 3,1)=(-6,1) \)
• Stretch vertically by 4: \( (-6,1\times4)=(-6,4) \)

1. For the point \( (x,y)=(\frac{1}{3},-1) \) on \( y=\log_{3}(x) \):

• Reflect over \( y \)-axis: \( (-\frac{1}{3},-1) \)
• Shift left 3 units: \( (-\frac{1}{3}-3,-1)=(-\frac{10}{3},-1) \)
• Stretch vertically by 4: \( (-\frac{10}{3},-1\times4)=(-\frac{10}{3},-4) \)

To sketch the graph:

• Draw the vertical asymptote \( x=-3 \).
• Plot the key points \( (-4,0) \), \( (-6,4) \), \( (-\frac{10}{3},-4) \) (or approximate their positions).
• The graph should be a logarithmic curve with the vertical asymptote \( x=-3 \), passing through the plotted points, and having the shape of a reflected, shifted, and stretched logarithmic graph. The given graph in the problem is incorrect for \( y = 4\log_{3}(-x - 3) \) (since its asymptote and key points don't match our analysis). The correct graph should have a vertical asymptote at \( x=-3 \), be defined for \( x < - 3 \) (since \( -x - 3>0\Rightarrow x < - 3 \)), and have the shape consistent with the transformations we analyzed.

(Note: If the task was to identify the correct graph among options, the correct graph would have a vertical asymptote at \( x=-3 \), be on the left - hand side of \( x = - 3 \), and have points consistent with the transformed key points we found.)

If we were to just describe the process of sketching, the above steps outline how to transform the parent logarithmic function to get the graph of \( y = 4\log_{3}(-x - 3) \).