Question
Describe the difference(s) between the graph of f(x) = 1/x−2+3
and the graph of the parent function f(x) = 1/x
.
and the graph of the parent function f(x) = 1/x
.
Answers
GPT-4o mini
To describe the differences between the graph of \( f(x) = \frac{1}{x} - 2 + 3 \) and the parent function \( f(x) = \frac{1}{x} \), let's first simplify the given function:
1. **Simplification**:
\[
f(x) = \frac{1}{x} - 2 + 3 = \frac{1}{x} + 1
\]
Thus, we can rewrite the function as:
\[
f(x) = \frac{1}{x} + 1
\]
2. **Identifying Transformations**:
The transformation from the parent function \( f(x) = \frac{1}{x} \) to \( f(x) = \frac{1}{x} + 1 \) involves a vertical shift. Specifically, the entire graph of \( f(x) = \frac{1}{x} \) is shifted upward by 1 unit.
3. **Key Characteristics**:
- **Vertical Asymptote**: The vertical asymptote of \( f(x) = \frac{1}{x} \) is at \( x = 0 \). The vertical asymptote remains at \( x = 0 \) for \( f(x) = \frac{1}{x} + 1 \) since the shift does not affect the asymptote.
- **Horizontal Asymptote**: The horizontal asymptote of \( f(x) = \frac{1}{x} \) is \( y = 0 \). For \( f(x) = \frac{1}{x} + 1 \), the horizontal asymptote shifts to \( y = 1 \), indicating that as \( x \to \infty \) or \( x \to -\infty \), the function approaches 1.
- **Intercepts**: The parent function \( f(x) = \frac{1}{x} \) has no y-intercept (as \( f(0) \) is undefined) and approaches the x-axis. For \( f(x) = \frac{1}{x} + 1 \):
- There is still no x-intercept (since \( \frac{1}{x} + 1 = 0 \) leads to no valid solution).
- The y-intercept occurs at \( (0, 1) \) when you evaluate the function close to \( y = 1 \) for values around \( x = 0 \).
4. **Graph Behavior**:
- Both graphs exhibit similar behavior in the first and third quadrants, but the placement is shifted upward by 1 unit in the second and fourth quadrants.
- For large positive \( x \), \( f(x) = \frac{1}{x} + 1 \) approaches \( y = 1 \) from above. Similarly, for large negative \( x \), it approaches from below.
In summary, the primary difference between the graphs is the vertical shift of the entire graph of the parent function \( f(x) = \frac{1}{x} \) upward by 1 unit, resulting in the new horizontal asymptote \( y = 1 \).
1. **Simplification**:
\[
f(x) = \frac{1}{x} - 2 + 3 = \frac{1}{x} + 1
\]
Thus, we can rewrite the function as:
\[
f(x) = \frac{1}{x} + 1
\]
2. **Identifying Transformations**:
The transformation from the parent function \( f(x) = \frac{1}{x} \) to \( f(x) = \frac{1}{x} + 1 \) involves a vertical shift. Specifically, the entire graph of \( f(x) = \frac{1}{x} \) is shifted upward by 1 unit.
3. **Key Characteristics**:
- **Vertical Asymptote**: The vertical asymptote of \( f(x) = \frac{1}{x} \) is at \( x = 0 \). The vertical asymptote remains at \( x = 0 \) for \( f(x) = \frac{1}{x} + 1 \) since the shift does not affect the asymptote.
- **Horizontal Asymptote**: The horizontal asymptote of \( f(x) = \frac{1}{x} \) is \( y = 0 \). For \( f(x) = \frac{1}{x} + 1 \), the horizontal asymptote shifts to \( y = 1 \), indicating that as \( x \to \infty \) or \( x \to -\infty \), the function approaches 1.
- **Intercepts**: The parent function \( f(x) = \frac{1}{x} \) has no y-intercept (as \( f(0) \) is undefined) and approaches the x-axis. For \( f(x) = \frac{1}{x} + 1 \):
- There is still no x-intercept (since \( \frac{1}{x} + 1 = 0 \) leads to no valid solution).
- The y-intercept occurs at \( (0, 1) \) when you evaluate the function close to \( y = 1 \) for values around \( x = 0 \).
4. **Graph Behavior**:
- Both graphs exhibit similar behavior in the first and third quadrants, but the placement is shifted upward by 1 unit in the second and fourth quadrants.
- For large positive \( x \), \( f(x) = \frac{1}{x} + 1 \) approaches \( y = 1 \) from above. Similarly, for large negative \( x \), it approaches from below.
In summary, the primary difference between the graphs is the vertical shift of the entire graph of the parent function \( f(x) = \frac{1}{x} \) upward by 1 unit, resulting in the new horizontal asymptote \( y = 1 \).