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Sure, let's go through the steps to plot the graph of the function [tex]\( f(x) = 3^{x-4} - 5 \)[/tex].
### Step 1: Understand the Function
The function [tex]\( f(x) = 3^{x-4} - 5 \)[/tex] consists of two parts:
1. The exponential component [tex]\( 3^{x-4} \)[/tex]
2. A vertical shift downward by 5 units (the [tex]\(-5\)[/tex])
### Step 2: Key Points and Features
1. Horizontal Shift: The term [tex]\( x-4 \)[/tex] in the exponent shifts the basic function [tex]\( 3^x \)[/tex] to the right by 4 units.
2. Vertical Shift: The [tex]\(-5\)[/tex] subtracts 5 from the function's value, shifting the entire graph downward by 5 units.
3. Asymptote: The horizontal asymptote originally for [tex]\( 3^x \)[/tex] is [tex]\( y=0 \)[/tex]. For [tex]\( f(x) \)[/tex], it shifts downward to [tex]\( y=-5 \)[/tex] because of the [tex]\(-5\)[/tex] in the function.
4. Intercepts:
- y-intercept: Evaluate [tex]\( f(0) \)[/tex]. Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3^{0-4} - 5 = 3^{-4} - 5 = \left(\frac{1}{81}\right) - 5 \approx -4.98765 \][/tex]
- x-intercept: Solve the equation [tex]\( 3^{x-4} - 5 = 0 \)[/tex]:
[tex]\[ 3^{x-4} = 5 \][/tex]
Taking the natural logarithm on both sides:
[tex]\[ \ln(3^{x-4}) = \ln(5) \][/tex]
Using properties of logarithms, we get:
[tex]\[ (x-4) \ln(3) = \ln(5) \][/tex]
Therefore:
[tex]\[ x-4 = \frac{\ln(5)}{\ln(3)} \][/tex]
[tex]\[ x = 4 + \frac{\ln(5)}{\ln(3)} \][/tex]
Numerically:
[tex]\[ x \approx 4 + 1.46497 \approx 5.46497 \][/tex]
### Step 3: Plot Points and Sketch
1. Create a Table of Values:
To accurately plot the graph, pick a few values for [tex]\( x \)[/tex] around the key points like 0, 4, and 5.46497.
| x | [tex]\( f(x) \)[/tex] |
|-------|---------------|
| 0 | -4.98765 |
| 2 | -4.88888 |
| 4 | -4 |
| 6 | -2.565 |
| 8 | 3.444 |
2. Draw the Graph:
- Plot the points from the table above.
- Draw the horizontal asymptote at [tex]\( y = -5 \)[/tex].
- Sketch the graph making sure it approaches the asymptote as [tex]\( x \)[/tex] goes to negative infinity.
- The graph should steeply rise after the x-intercept [tex]\( x \approx 5.46497 \)[/tex].
### Final Graph:
Your graph should visually depict the horizontal asymptote at [tex]\( y = -5 \)[/tex], a rightward shift, and the features discussed above. Since we can't literally draw it here, these steps should guide you to sketch the graph accurately on graph paper or using graphing software.
### Step 1: Understand the Function
The function [tex]\( f(x) = 3^{x-4} - 5 \)[/tex] consists of two parts:
1. The exponential component [tex]\( 3^{x-4} \)[/tex]
2. A vertical shift downward by 5 units (the [tex]\(-5\)[/tex])
### Step 2: Key Points and Features
1. Horizontal Shift: The term [tex]\( x-4 \)[/tex] in the exponent shifts the basic function [tex]\( 3^x \)[/tex] to the right by 4 units.
2. Vertical Shift: The [tex]\(-5\)[/tex] subtracts 5 from the function's value, shifting the entire graph downward by 5 units.
3. Asymptote: The horizontal asymptote originally for [tex]\( 3^x \)[/tex] is [tex]\( y=0 \)[/tex]. For [tex]\( f(x) \)[/tex], it shifts downward to [tex]\( y=-5 \)[/tex] because of the [tex]\(-5\)[/tex] in the function.
4. Intercepts:
- y-intercept: Evaluate [tex]\( f(0) \)[/tex]. Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3^{0-4} - 5 = 3^{-4} - 5 = \left(\frac{1}{81}\right) - 5 \approx -4.98765 \][/tex]
- x-intercept: Solve the equation [tex]\( 3^{x-4} - 5 = 0 \)[/tex]:
[tex]\[ 3^{x-4} = 5 \][/tex]
Taking the natural logarithm on both sides:
[tex]\[ \ln(3^{x-4}) = \ln(5) \][/tex]
Using properties of logarithms, we get:
[tex]\[ (x-4) \ln(3) = \ln(5) \][/tex]
Therefore:
[tex]\[ x-4 = \frac{\ln(5)}{\ln(3)} \][/tex]
[tex]\[ x = 4 + \frac{\ln(5)}{\ln(3)} \][/tex]
Numerically:
[tex]\[ x \approx 4 + 1.46497 \approx 5.46497 \][/tex]
### Step 3: Plot Points and Sketch
1. Create a Table of Values:
To accurately plot the graph, pick a few values for [tex]\( x \)[/tex] around the key points like 0, 4, and 5.46497.
| x | [tex]\( f(x) \)[/tex] |
|-------|---------------|
| 0 | -4.98765 |
| 2 | -4.88888 |
| 4 | -4 |
| 6 | -2.565 |
| 8 | 3.444 |
2. Draw the Graph:
- Plot the points from the table above.
- Draw the horizontal asymptote at [tex]\( y = -5 \)[/tex].
- Sketch the graph making sure it approaches the asymptote as [tex]\( x \)[/tex] goes to negative infinity.
- The graph should steeply rise after the x-intercept [tex]\( x \approx 5.46497 \)[/tex].
### Final Graph:
Your graph should visually depict the horizontal asymptote at [tex]\( y = -5 \)[/tex], a rightward shift, and the features discussed above. Since we can't literally draw it here, these steps should guide you to sketch the graph accurately on graph paper or using graphing software.
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