Answer :
To graph the function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex] and to plot two points with integer coordinates, let's follow these steps in detail:
### Step-by-Step Solution
1. Understanding the Function:
[tex]\[ y = -\log_{\frac{1}{6}}(x) \][/tex]
This equation transforms the logarithm from base [tex]\(\frac{1}{6}\)[/tex] by negating it, which affects the shape and direction of the graph.
2. Change of Base Formula:
To work with logarithms of any base, we use the change of base formula:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{\log(\frac{1}{6})} \][/tex]
Given that [tex]\(\log(\frac{1}{6}) = \log(1) - \log(6) = 0 - \log(6) = -\log(6)\)[/tex], we can rewrite:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{-\log(6)} = -\frac{\log(x)}{\log(6)} \][/tex]
Hence, our function becomes:
[tex]\[ y = -\left( -\frac{\log(x)}{\log(6)} \right) = \frac{\log(x)}{\log(6)} \][/tex]
3. Plotting the Graph:
We will plot the function:
[tex]\[ y = \frac{\log(x)}{\log(6)} \][/tex]
4. Calculate Specific Points:
To plot points with integer coordinates, we need specific [tex]\( x \)[/tex] values where both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are integers.
- First Point:
Let’s try [tex]\( x = 6 \)[/tex]:
[tex]\[ y = \frac{\log(6)}{\log(6)} = 1 \][/tex]
The first point is [tex]\( (6, 1) \)[/tex].
- Second Point:
Let’s try [tex]\( x = 36 \)[/tex]:
[tex]\[ y = \frac{\log(36)}{\log(6)} = \frac{2 \log(6)}{\log(6)} = 2 \][/tex]
The second point is [tex]\( (36, 2) \)[/tex], but since [tex]\( x = 36 \)[/tex] is not practical for visualization, we try another.
Alternatively, let’s use [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \frac{\log(1)}{\log(6)} = \frac{0}{\log(6)} = 0 \][/tex]
The second point is [tex]\( (1, 0) \)[/tex].
### Final Graph
- Plot the Points:
[tex]\[ (6, 1) \quad \text{and} \quad (1, 0) \][/tex]
- Plot the Function:
The curve of [tex]\( y = \frac{\log(x)}{\log(6)} \)[/tex] (which is equivalent to the original function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]) is shown on a standard [tex]\( xy \)[/tex]-coordinate plane.
- Shape and Direction:
Given the nature of logarithmic functions, particularly with this transformation, the graph will pass through the points and exhibit the typical increasing logarithmic behavior, rising more slowly as [tex]\( x \)[/tex] increases.
To finalize, you should plot the points [tex]\((6, 1)\)[/tex] and [tex]\((1, 0)\)[/tex] on the [tex]\( xy \)[/tex]-plane and draw a smooth curve through these points to represent the function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]. The graph will start from [tex]\((1, 0)\)[/tex] and pass through [tex]\((6, 1)\)[/tex], continuing to increase gradually as [tex]\( x \)[/tex] increases.
### Step-by-Step Solution
1. Understanding the Function:
[tex]\[ y = -\log_{\frac{1}{6}}(x) \][/tex]
This equation transforms the logarithm from base [tex]\(\frac{1}{6}\)[/tex] by negating it, which affects the shape and direction of the graph.
2. Change of Base Formula:
To work with logarithms of any base, we use the change of base formula:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{\log(\frac{1}{6})} \][/tex]
Given that [tex]\(\log(\frac{1}{6}) = \log(1) - \log(6) = 0 - \log(6) = -\log(6)\)[/tex], we can rewrite:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{-\log(6)} = -\frac{\log(x)}{\log(6)} \][/tex]
Hence, our function becomes:
[tex]\[ y = -\left( -\frac{\log(x)}{\log(6)} \right) = \frac{\log(x)}{\log(6)} \][/tex]
3. Plotting the Graph:
We will plot the function:
[tex]\[ y = \frac{\log(x)}{\log(6)} \][/tex]
4. Calculate Specific Points:
To plot points with integer coordinates, we need specific [tex]\( x \)[/tex] values where both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are integers.
- First Point:
Let’s try [tex]\( x = 6 \)[/tex]:
[tex]\[ y = \frac{\log(6)}{\log(6)} = 1 \][/tex]
The first point is [tex]\( (6, 1) \)[/tex].
- Second Point:
Let’s try [tex]\( x = 36 \)[/tex]:
[tex]\[ y = \frac{\log(36)}{\log(6)} = \frac{2 \log(6)}{\log(6)} = 2 \][/tex]
The second point is [tex]\( (36, 2) \)[/tex], but since [tex]\( x = 36 \)[/tex] is not practical for visualization, we try another.
Alternatively, let’s use [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \frac{\log(1)}{\log(6)} = \frac{0}{\log(6)} = 0 \][/tex]
The second point is [tex]\( (1, 0) \)[/tex].
### Final Graph
- Plot the Points:
[tex]\[ (6, 1) \quad \text{and} \quad (1, 0) \][/tex]
- Plot the Function:
The curve of [tex]\( y = \frac{\log(x)}{\log(6)} \)[/tex] (which is equivalent to the original function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]) is shown on a standard [tex]\( xy \)[/tex]-coordinate plane.
- Shape and Direction:
Given the nature of logarithmic functions, particularly with this transformation, the graph will pass through the points and exhibit the typical increasing logarithmic behavior, rising more slowly as [tex]\( x \)[/tex] increases.
To finalize, you should plot the points [tex]\((6, 1)\)[/tex] and [tex]\((1, 0)\)[/tex] on the [tex]\( xy \)[/tex]-plane and draw a smooth curve through these points to represent the function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]. The graph will start from [tex]\((1, 0)\)[/tex] and pass through [tex]\((6, 1)\)[/tex], continuing to increase gradually as [tex]\( x \)[/tex] increases.
