Question
a reciprocal function is shown below:
y = (1/x)
How can I use its graph to prove that the equation is y=(1/x)
y = (1/x)
How can I use its graph to prove that the equation is y=(1/x)
Answers
GPT-4o mini
To prove that a graph represents the reciprocal function \( y = \frac{1}{x} \), you can follow these steps:
1. **Identify Key Features of the Graph:**
- The graph of \( y = \frac{1}{x} \) has two branches: one in the first quadrant and one in the third quadrant.
- The function is not defined at \( x = 0 \) (there is a vertical asymptote at \( x = 0 \)).
- As \( x \) approaches 0 from the right (\( x \to 0^+ \)), \( y \) approaches infinity (\( y \to \infty \)).
- As \( x \) approaches 0 from the left (\( x \to 0^- \)), \( y \) approaches negative infinity (\( y \to -\infty \)).
- The graph approaches the x-axis as \( x \) goes to positive or negative infinity (horizontal asymptote at \( y = 0 \)).
2. **Examine Symmetry:**
- Verify that the graph is symmetric about the origin. This means that if the point \( (a, b) \) is on the graph, then the point \( (-a, -b) \) should also be on the graph. For \( y = \frac{1}{x} \), if \( x = a \), then \( y = \frac{1}{a} \), and for \( x = -a \), \( y = \frac{1}{-a} = -\frac{1}{a} \).
- This demonstrates that \( y = \frac{1}{x} \) has odd symmetry.
3. **Plot Specific Points:**
- Choose specific values for \( x \) and calculate corresponding \( y \) values (e.g., \( x = 1 \Rightarrow y = 1 \); \( x = 2 \Rightarrow y = 0.5 \); \( x = -1 \Rightarrow y = -1 \); \( x = -2 \Rightarrow y = -0.5 \)).
- Plot these points and ensure they align with the graph.
4. **Behavior at Extremes:**
- Analyze the behavior of the function as \( x \) approaches positive and negative infinity. You will notice that \( y \) approaches 0 but never actually reaches 0, confirming that the graph agrees with \( y = \frac{1}{x} \).
5. **Conclude:**
- If all the characteristics match up with the expected properties of the function \( y = \frac{1}{x} \), then you have successfully proved that the graph represents the reciprocal function \( y = \frac{1}{x} \).
By following this approach, you can demonstrate that the behavior and features of the graph correspond to the function \( y = \frac{1}{x} \).
1. **Identify Key Features of the Graph:**
- The graph of \( y = \frac{1}{x} \) has two branches: one in the first quadrant and one in the third quadrant.
- The function is not defined at \( x = 0 \) (there is a vertical asymptote at \( x = 0 \)).
- As \( x \) approaches 0 from the right (\( x \to 0^+ \)), \( y \) approaches infinity (\( y \to \infty \)).
- As \( x \) approaches 0 from the left (\( x \to 0^- \)), \( y \) approaches negative infinity (\( y \to -\infty \)).
- The graph approaches the x-axis as \( x \) goes to positive or negative infinity (horizontal asymptote at \( y = 0 \)).
2. **Examine Symmetry:**
- Verify that the graph is symmetric about the origin. This means that if the point \( (a, b) \) is on the graph, then the point \( (-a, -b) \) should also be on the graph. For \( y = \frac{1}{x} \), if \( x = a \), then \( y = \frac{1}{a} \), and for \( x = -a \), \( y = \frac{1}{-a} = -\frac{1}{a} \).
- This demonstrates that \( y = \frac{1}{x} \) has odd symmetry.
3. **Plot Specific Points:**
- Choose specific values for \( x \) and calculate corresponding \( y \) values (e.g., \( x = 1 \Rightarrow y = 1 \); \( x = 2 \Rightarrow y = 0.5 \); \( x = -1 \Rightarrow y = -1 \); \( x = -2 \Rightarrow y = -0.5 \)).
- Plot these points and ensure they align with the graph.
4. **Behavior at Extremes:**
- Analyze the behavior of the function as \( x \) approaches positive and negative infinity. You will notice that \( y \) approaches 0 but never actually reaches 0, confirming that the graph agrees with \( y = \frac{1}{x} \).
5. **Conclude:**
- If all the characteristics match up with the expected properties of the function \( y = \frac{1}{x} \), then you have successfully proved that the graph represents the reciprocal function \( y = \frac{1}{x} \).
By following this approach, you can demonstrate that the behavior and features of the graph correspond to the function \( y = \frac{1}{x} \).