Answer :
To graph the function [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex], follow these detailed steps:
### 1. Understanding the Function
The given function [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex] is an exponential function, where the base of the exponential term is [tex]\( 2 \)[/tex]. This type of function grows (or decays) exponentially as [tex]\( x \)[/tex] increases (or decreases).
### 2. Choosing Points to Plot
To effectively graph the function, let's choose several [tex]\( x \)[/tex]-values to calculate the corresponding [tex]\( g(x) \)[/tex]-values.
#### Selected [tex]\( x \)[/tex]-values:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = -2 \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = 3 \)[/tex]
#### Corresponding [tex]\( g(x) \)[/tex]-values:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ g(-3) = \frac{3}{4} \cdot 2^{-3} = \frac{3}{4} \cdot \frac{1}{8} = \frac{3}{32} \approx 0.094 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = \frac{3}{4} \cdot 2^{-2} = \frac{3}{4} \cdot \frac{1}{4} = \frac{3}{16} = 0.1875 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = \frac{3}{4} \cdot 2^{-1} = \frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8} = 0.375 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = \frac{3}{4} \cdot 2^0 = \frac{3}{4} \cdot 1 = \frac{3}{4} = 0.75 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = \frac{3}{4} \cdot 2^1 = \frac{3}{4} \cdot 2 = \frac{3 \cdot 2}{4} = \frac{6}{4} = 1.5 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = \frac{3}{4} \cdot 2^2 = \frac{3}{4} \cdot 4 = \frac{3 \cdot 4}{4} = 3 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = \frac{3}{4} \cdot 2^3 = \frac{3}{4} \cdot 8 = \frac{3 \cdot 8}{4} = 6 \][/tex]
### 3. Plotting the Points
Using the points calculated above, we can plot the function manually on a graph. For a more complete and smooth curve, we should connect those points.
### 4. Drawing the Axes
- Draw a horizontal (x-axis) and a vertical (y-axis) line on graph paper or using graphing software.
- Mark the x-axis with values: [tex]\( -3, -2, -1, 0, 1, 2, 3 \)[/tex].
- Mark the y-axis with values to adequately capture the range of [tex]\( g(x) \)[/tex]-values calculated: [tex]\( 0, 1, 2, 3, 4, 5, 6 \)[/tex].
### 5. Plotting Points and Drawing the Curve
- Plot the points [tex]\((-3, 0.094)\)[/tex], [tex]\((-2, 0.1875)\)[/tex], [tex]\((-1, 0.375)\)[/tex], [tex]\((0, 0.75)\)[/tex], [tex]\((1, 1.5)\)[/tex], [tex]\((2, 3)\)[/tex], [tex]\((3, 6)\)[/tex].
- After plotting these points, draw a smooth curve through them to represent the exponential growth of [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex].
### 6. Labeling the Graph
- Title the graph "Graph of the Function [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex]".
- Label the x-axis with "x".
- Label the y-axis with "g(x)".
By following these steps, you can accurately graph the exponential function [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex], showing its behavior as [tex]\( x \)[/tex] varies from [tex]\( -3 \)[/tex] to [tex]\( 3 \)[/tex] and beyond.
### 1. Understanding the Function
The given function [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex] is an exponential function, where the base of the exponential term is [tex]\( 2 \)[/tex]. This type of function grows (or decays) exponentially as [tex]\( x \)[/tex] increases (or decreases).
### 2. Choosing Points to Plot
To effectively graph the function, let's choose several [tex]\( x \)[/tex]-values to calculate the corresponding [tex]\( g(x) \)[/tex]-values.
#### Selected [tex]\( x \)[/tex]-values:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = -2 \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = 3 \)[/tex]
#### Corresponding [tex]\( g(x) \)[/tex]-values:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ g(-3) = \frac{3}{4} \cdot 2^{-3} = \frac{3}{4} \cdot \frac{1}{8} = \frac{3}{32} \approx 0.094 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = \frac{3}{4} \cdot 2^{-2} = \frac{3}{4} \cdot \frac{1}{4} = \frac{3}{16} = 0.1875 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = \frac{3}{4} \cdot 2^{-1} = \frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8} = 0.375 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = \frac{3}{4} \cdot 2^0 = \frac{3}{4} \cdot 1 = \frac{3}{4} = 0.75 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = \frac{3}{4} \cdot 2^1 = \frac{3}{4} \cdot 2 = \frac{3 \cdot 2}{4} = \frac{6}{4} = 1.5 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = \frac{3}{4} \cdot 2^2 = \frac{3}{4} \cdot 4 = \frac{3 \cdot 4}{4} = 3 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = \frac{3}{4} \cdot 2^3 = \frac{3}{4} \cdot 8 = \frac{3 \cdot 8}{4} = 6 \][/tex]
### 3. Plotting the Points
Using the points calculated above, we can plot the function manually on a graph. For a more complete and smooth curve, we should connect those points.
### 4. Drawing the Axes
- Draw a horizontal (x-axis) and a vertical (y-axis) line on graph paper or using graphing software.
- Mark the x-axis with values: [tex]\( -3, -2, -1, 0, 1, 2, 3 \)[/tex].
- Mark the y-axis with values to adequately capture the range of [tex]\( g(x) \)[/tex]-values calculated: [tex]\( 0, 1, 2, 3, 4, 5, 6 \)[/tex].
### 5. Plotting Points and Drawing the Curve
- Plot the points [tex]\((-3, 0.094)\)[/tex], [tex]\((-2, 0.1875)\)[/tex], [tex]\((-1, 0.375)\)[/tex], [tex]\((0, 0.75)\)[/tex], [tex]\((1, 1.5)\)[/tex], [tex]\((2, 3)\)[/tex], [tex]\((3, 6)\)[/tex].
- After plotting these points, draw a smooth curve through them to represent the exponential growth of [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex].
### 6. Labeling the Graph
- Title the graph "Graph of the Function [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex]".
- Label the x-axis with "x".
- Label the y-axis with "g(x)".
By following these steps, you can accurately graph the exponential function [tex]\( g(x) = \frac{3}{4} \cdot 2^x \)[/tex], showing its behavior as [tex]\( x \)[/tex] varies from [tex]\( -3 \)[/tex] to [tex]\( 3 \)[/tex] and beyond.
