Answer :
To draw the graph of the function [tex]\( f(x) = \left( \frac{1}{6} \right)^{x-1} \)[/tex], let's break down the steps and understand the behavior of the function.
### Step-by-Step Solution:
1. Understand the Function:
The function [tex]\( f(x) = \left( \frac{1}{6} \right)^{x-1} \)[/tex] is an exponential function where the base [tex]\( \frac{1}{6} \)[/tex] is less than 1. Exponential functions of this form are decreasing functions.
2. Transformations:
The expression [tex]\( (x - 1) \)[/tex] in the exponent shifts the basic function [tex]\( \left( \frac{1}{6} \right)^x \)[/tex] horizontally to the right by 1 unit. This means instead of starting at [tex]\( x = 0 \)[/tex], the significant behavior starts at [tex]\( x = 1 \)[/tex].
3. Key Points:
Let's calculate a few key points to aid in sketching the graph:
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \left( \frac{1}{6} \right)^{1-1} = \left( \frac{1}{6} \right)^0 = 1 \][/tex]
- At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \left( \frac{1}{6} \right)^{2-1} = \left( \frac{1}{6} \right)^1 = \frac{1}{6} \][/tex]
- At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \left( \frac{1}{6} \right)^{3-1} = \left( \frac{1}{6} \right)^2 = \left( \frac{1}{6} \right) \left( \frac{1}{6} \right) = \frac{1}{36} \][/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \left( \frac{1}{6} \right)^{0-1} = \left( \frac{1}{6} \right)^{-1} = 6 \][/tex]
Since the function decreases rapidly as [tex]\( x \)[/tex] increases, for large values of [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] will approach zero.
4. Behavior Asymptotically:
As [tex]\( x \)[/tex] goes to positive infinity ([tex]\( x \to +\infty \)[/tex]), [tex]\( f(x) \)[/tex] approaches zero. As [tex]\( x \)[/tex] goes to negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( f(x) \)[/tex] will increase without bound since the fraction to a negative power results in a large value.
5. Graph Sketching:
- The point [tex]\((0, 6)\)[/tex] shows where the function starts.
- The point [tex]\((1, 1)\)[/tex] is crucial because it is an easy-to-calculate point where the function equals 1.
- The function will be decreasing because [tex]\( \frac{1}{6} \)[/tex] is less than 1.
- As [tex]\( x \)[/tex] becomes more positive, [tex]\( f(x) \)[/tex] gets closer and closer to 0.
- For negative [tex]\( x \)[/tex], as it becomes more negative, the function grows rapidly.
### Graph:
To sketch the graph:
1. Draw the axes and label them (x-axis and y-axis).
2. Plot the key points: [tex]\((0, 6)\)[/tex], [tex]\((1, 1)\)[/tex], [tex]\((2, \frac{1}{6})\)[/tex], and [tex]\((3, \frac{1}{36})\)[/tex].
3. Draw a smooth curve that passes through the plotted points.
4. Ensure that as [tex]\( x \)[/tex] increases, the curve approaches 0 but never actually touches the x-axis (asymptote).
5. As [tex]\( x \)[/tex] moves to the left, the function increases rapidly.
The graph of [tex]\( f(x) = \left( \frac{1}{6} \right)^{x-1} \)[/tex] is shown below:
[tex]\[ \begin{array}{ccc} y & & \\ | & & \\ 6 & \cdots & \text{{Point }} (0,6) \\ | & & \\ 1 & \cdot & \text{{Point }} (1,1) \\ | & & \\ | & & \\ | & \cdot & \text{{Point}} (2, \frac{1}{6}) \\ | & & \\ | & & \\ | & & \\ \cdots & & x \\ \end{array} \][/tex]
Draw a smooth curve connecting these points, considering the asymptotic behavior described.
### Step-by-Step Solution:
1. Understand the Function:
The function [tex]\( f(x) = \left( \frac{1}{6} \right)^{x-1} \)[/tex] is an exponential function where the base [tex]\( \frac{1}{6} \)[/tex] is less than 1. Exponential functions of this form are decreasing functions.
2. Transformations:
The expression [tex]\( (x - 1) \)[/tex] in the exponent shifts the basic function [tex]\( \left( \frac{1}{6} \right)^x \)[/tex] horizontally to the right by 1 unit. This means instead of starting at [tex]\( x = 0 \)[/tex], the significant behavior starts at [tex]\( x = 1 \)[/tex].
3. Key Points:
Let's calculate a few key points to aid in sketching the graph:
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \left( \frac{1}{6} \right)^{1-1} = \left( \frac{1}{6} \right)^0 = 1 \][/tex]
- At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \left( \frac{1}{6} \right)^{2-1} = \left( \frac{1}{6} \right)^1 = \frac{1}{6} \][/tex]
- At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \left( \frac{1}{6} \right)^{3-1} = \left( \frac{1}{6} \right)^2 = \left( \frac{1}{6} \right) \left( \frac{1}{6} \right) = \frac{1}{36} \][/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \left( \frac{1}{6} \right)^{0-1} = \left( \frac{1}{6} \right)^{-1} = 6 \][/tex]
Since the function decreases rapidly as [tex]\( x \)[/tex] increases, for large values of [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] will approach zero.
4. Behavior Asymptotically:
As [tex]\( x \)[/tex] goes to positive infinity ([tex]\( x \to +\infty \)[/tex]), [tex]\( f(x) \)[/tex] approaches zero. As [tex]\( x \)[/tex] goes to negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( f(x) \)[/tex] will increase without bound since the fraction to a negative power results in a large value.
5. Graph Sketching:
- The point [tex]\((0, 6)\)[/tex] shows where the function starts.
- The point [tex]\((1, 1)\)[/tex] is crucial because it is an easy-to-calculate point where the function equals 1.
- The function will be decreasing because [tex]\( \frac{1}{6} \)[/tex] is less than 1.
- As [tex]\( x \)[/tex] becomes more positive, [tex]\( f(x) \)[/tex] gets closer and closer to 0.
- For negative [tex]\( x \)[/tex], as it becomes more negative, the function grows rapidly.
### Graph:
To sketch the graph:
1. Draw the axes and label them (x-axis and y-axis).
2. Plot the key points: [tex]\((0, 6)\)[/tex], [tex]\((1, 1)\)[/tex], [tex]\((2, \frac{1}{6})\)[/tex], and [tex]\((3, \frac{1}{36})\)[/tex].
3. Draw a smooth curve that passes through the plotted points.
4. Ensure that as [tex]\( x \)[/tex] increases, the curve approaches 0 but never actually touches the x-axis (asymptote).
5. As [tex]\( x \)[/tex] moves to the left, the function increases rapidly.
The graph of [tex]\( f(x) = \left( \frac{1}{6} \right)^{x-1} \)[/tex] is shown below:
[tex]\[ \begin{array}{ccc} y & & \\ | & & \\ 6 & \cdots & \text{{Point }} (0,6) \\ | & & \\ 1 & \cdot & \text{{Point }} (1,1) \\ | & & \\ | & & \\ | & \cdot & \text{{Point}} (2, \frac{1}{6}) \\ | & & \\ | & & \\ | & & \\ \cdots & & x \\ \end{array} \][/tex]
Draw a smooth curve connecting these points, considering the asymptotic behavior described.
