Answer
**(a) The graph of \( f(x) = 2^{x} - 4 \) is an exponential curve shifted downward by 4 units, passing through the point \( (2, 0) \).
**(b) The zero of \( f(x) \) is at \( x = 2 \).
**(c) \( f(x) < 0 \) for all real numbers \( x \) where \( x < 2 \).
Solution
Sure, let's tackle each part of the problem step by step.
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### **(a) Graph of \( f(x) = 2^{x} - 4 \)**
To graph the function \( f(x) = 2^{x} - 4 \), we'll analyze its key features:
1. **Base Function**: The base exponential function is \( 2^{x} \), which has the following characteristics:
- **Y-intercept**: \( (0, 1) \) because \( 2^{0} = 1 \).
- **Horizontal Asymptote**: \( y = 0 \) as \( x \to -\infty \).
- **Growth**: Increases rapidly as \( x \) increases.
2. **Transformation**: The function \( 2^{x} - 4 \) shifts the base function **downward** by 4 units.
- **New Y-intercept**: \( (0, 1 - 4) = (0, -3) \).
- **Horizontal Asymptote**: \( y = -4 \).
3. **Plotting Points**:
| \( x \) | \( 2^{x} \) | \( f(x) = 2^{x} - 4 \) |
|---------|------------|------------------------|
| -2 | 0.25 | -3.75 |
| -1 | 0.5 | -3.5 |
| 0 | 1 | -3 |
| 1 | 2 | -2 |
| 2 | 4 | 0 |
| 3 | 8 | 4 |
| 4 | 16 | 12 |
4. **Graph Description**:
- Starts near the horizontal asymptote \( y = -4 \) for large negative \( x \) values.
- Passes through the y-intercept at \( (0, -3) \).
- Crosses the x-axis at \( x = 2 \) (zero of the function).
- Increases rapidly as \( x \) becomes positive.
*Since this is a text-based medium, please sketch the graph using the points above, ensuring the exponential growth and downward shift are represented.*
---
### **(b) Finding the Zero of \( f(x) \)**
To find the zero of the function \( f(x) = 2^{x} - 4 \), set \( f(x) \) equal to zero and solve for \( x \):
\[
2^{x} - 4 = 0 \\
2^{x} = 4
\]
Since \( 4 \) is a power of \( 2 \), we can write:
\[
2^{x} = 2^{2} \\
x = 2
\]
**Zero of \( f(x) \):** \( x = 2 \)
*This means the graph of \( f(x) = 2^{x} - 4 \) crosses the x-axis at the point \( (2, 0) \).*
---
### **(c) Solving \( f(x) < 0 \) Based on the Graph**
We need to determine the values of \( x \) for which \( f(x) = 2^{x} - 4 \) is less than zero:
\[
2^{x} - 4 < 0 \\
2^{x} < 4
\]
Again, recognizing that \( 4 = 2^{2} \):
\[
2^{x} < 2^{2}
\]
Since the base \( 2 \) is greater than 1, the inequality holds when the exponent is less than 2:
\[
x < 2
\]
**Solution:** All real numbers \( x \) such that \( x < 2 \).
*Graphically, this corresponds to the portion of the graph below the x-axis, which occurs to the left of the vertical line \( x = 2 \).*
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### **Summary**
- **(a)** The graph of \( f(x) = 2^{x} - 4 \) is an exponential curve shifted downward by 4 units, with a y-intercept at \( (0, -3) \), a horizontal asymptote at \( y = -4 \), and passing through \( (2, 0) \).
- **(b)** The zero of \( f(x) \) is at \( x = 2 \).
- **(c)** \( f(x) < 0 \) for all real numbers \( x \) where \( x < 2 \).
Reviewed and approved by the UpStudy tutoring team
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