Question
Describe the difference between the graphs of f(x) = 1/(x - 2) + 3 the parent function f(x) = 1/x
Answers
Answered by
GPT-4o mini
To compare the graphs of the function \( f(x) = \frac{1}{x - 2} + 3 \) with its parent function \( g(x) = \frac{1}{x} \), we need to analyze the transformations that have been applied to the parent function.
### Parent Function: \( g(x) = \frac{1}{x} \)
1. **Vertical Asymptote**: The graph has a vertical asymptote at \( x = 0 \).
2. **Horizontal Asymptote**: The graph has a horizontal asymptote at \( y = 0 \).
3. **Behavior**: As \( x \) approaches 0 from the left, \( g(x) \) approaches \(-\infty\), and as \( x \) approaches 0 from the right, \( g(x) \) approaches \(+\infty\). For large positive or negative \( x \), \( g(x) \) approaches 0.
### Transformed Function: \( f(x) = \frac{1}{x - 2} + 3 \)
To understand the differences, let's break down the transformations:
1. **Horizontal Shift**: The term \( (x - 2) \) indicates a horizontal shift to the right by 2 units. Therefore, the vertical asymptote of \( f(x) \) is now at \( x = 2 \) instead of \( x = 0 \).
2. **Vertical Shift**: The addition of 3 shifts the entire graph upwards by 3 units, so the horizontal asymptote of \( f(x) \) is now at \( y = 3 \) instead of \( y = 0 \).
### Summary of Differences:
- **Vertical Asymptote**:
- \( g(x) \): \( x = 0 \)
- \( f(x) \): \( x = 2 \)
- **Horizontal Asymptote**:
- \( g(x) \): \( y = 0 \)
- \( f(x) \): \( y = 3 \)
- **Graph Behavior**:
- For \( g(x) \), as \( x \) gets large (positive or negative), \( g(x) \) approaches 0.
- For \( f(x) \), as \( x \) gets large (positive), \( f(x) \) approaches 3, and as \( x \) approaches 2 from the left, \( f(x) \) approaches \( -\infty \), while from the right, it approaches \( +\infty \).
### Graph Sketch:
- The general shape of both graphs is similar, resembling branches of a hyperbola. However, the branches of \( f(x) \) will be located in different quadrants shifted due to the transformations:
- The left branch (for \( x < 2 \)) will start high above \( y = 3 \) and drop down to \( -\infty \) as \( x \) approaches 2.
- The right branch (for \( x > 2 \)) will start just above \( y = 3 \) and approach the horizontal asymptote of \( y = 3 \) as \( x \) increases.
These transformations result in a graph that is similar in shape to that of the parent function but located differently on the coordinate plane.
### Parent Function: \( g(x) = \frac{1}{x} \)
1. **Vertical Asymptote**: The graph has a vertical asymptote at \( x = 0 \).
2. **Horizontal Asymptote**: The graph has a horizontal asymptote at \( y = 0 \).
3. **Behavior**: As \( x \) approaches 0 from the left, \( g(x) \) approaches \(-\infty\), and as \( x \) approaches 0 from the right, \( g(x) \) approaches \(+\infty\). For large positive or negative \( x \), \( g(x) \) approaches 0.
### Transformed Function: \( f(x) = \frac{1}{x - 2} + 3 \)
To understand the differences, let's break down the transformations:
1. **Horizontal Shift**: The term \( (x - 2) \) indicates a horizontal shift to the right by 2 units. Therefore, the vertical asymptote of \( f(x) \) is now at \( x = 2 \) instead of \( x = 0 \).
2. **Vertical Shift**: The addition of 3 shifts the entire graph upwards by 3 units, so the horizontal asymptote of \( f(x) \) is now at \( y = 3 \) instead of \( y = 0 \).
### Summary of Differences:
- **Vertical Asymptote**:
- \( g(x) \): \( x = 0 \)
- \( f(x) \): \( x = 2 \)
- **Horizontal Asymptote**:
- \( g(x) \): \( y = 0 \)
- \( f(x) \): \( y = 3 \)
- **Graph Behavior**:
- For \( g(x) \), as \( x \) gets large (positive or negative), \( g(x) \) approaches 0.
- For \( f(x) \), as \( x \) gets large (positive), \( f(x) \) approaches 3, and as \( x \) approaches 2 from the left, \( f(x) \) approaches \( -\infty \), while from the right, it approaches \( +\infty \).
### Graph Sketch:
- The general shape of both graphs is similar, resembling branches of a hyperbola. However, the branches of \( f(x) \) will be located in different quadrants shifted due to the transformations:
- The left branch (for \( x < 2 \)) will start high above \( y = 3 \) and drop down to \( -\infty \) as \( x \) approaches 2.
- The right branch (for \( x > 2 \)) will start just above \( y = 3 \) and approach the horizontal asymptote of \( y = 3 \) as \( x \) increases.
These transformations result in a graph that is similar in shape to that of the parent function but located differently on the coordinate plane.