Question
y-intercepts in Real-World Problems quick check
Answers
GPT-4o mini
Certainly! Understanding y-intercepts in real-world problems involves interpreting scenarios where you might have a relationship between two variables, typically in the form of a linear equation \(y = mx + b\), where \(b\) represents the y-intercept.
### Quick Check on y-Intercepts
1. **Understanding the Concept**:
- The y-intercept is the value of \(y\) when \(x = 0\).
- It often represents a starting point or initial condition in real-world contexts.
2. **Examples of Real-World Scenarios**:
- **Finance**: If you’re analyzing a savings account with a starting balance of $500 (the y-intercept), and you add a fixed amount every month, the y-intercept would be the initial amount without any additions.
- **Distance and Time**: In a situation where you’re measuring distance traveled over time, a scenario where you start at zero distance at time zero would have a y-intercept of zero. If you start with an initial distance of 10 km, the y-intercept would be 10.
3. **Identifying the y-Intercept**:
- Given an equation, determine if it's in slope-intercept form \(y = mx + b\). The \(b\) value is your y-intercept.
- In a graph, the y-intercept is where the line crosses the y-axis.
4. **Sample Problem**:
- **Problem Statement**: A car rental company charges a flat fee of $30 plus $0.20 per mile. Write the equation for the total cost \(C\) based on the number of miles \(m\).
- **Equation**: \(C = 0.20m + 30\)
- **y-Intercept**: When \(m = 0\) (no miles driven), \(C = 30\). Thus, the y-intercept is $30.
5. **Practice Questions**:
- What does the y-intercept represent in the context of a population growth model where \(y\) is the population and \(x\) is time (in years)?
- Given the equation \(y = 3x + 5\), identify the y-intercept and interpret its meaning.
### Conclusion
The y-intercept can provide crucial insights into a problem’s initial conditions and is essential for interpreting data in various fields like finance, biology, and physics. Understanding how to identify and interpret the y-intercept allows for better analysis of real-world situations.
### Quick Check on y-Intercepts
1. **Understanding the Concept**:
- The y-intercept is the value of \(y\) when \(x = 0\).
- It often represents a starting point or initial condition in real-world contexts.
2. **Examples of Real-World Scenarios**:
- **Finance**: If you’re analyzing a savings account with a starting balance of $500 (the y-intercept), and you add a fixed amount every month, the y-intercept would be the initial amount without any additions.
- **Distance and Time**: In a situation where you’re measuring distance traveled over time, a scenario where you start at zero distance at time zero would have a y-intercept of zero. If you start with an initial distance of 10 km, the y-intercept would be 10.
3. **Identifying the y-Intercept**:
- Given an equation, determine if it's in slope-intercept form \(y = mx + b\). The \(b\) value is your y-intercept.
- In a graph, the y-intercept is where the line crosses the y-axis.
4. **Sample Problem**:
- **Problem Statement**: A car rental company charges a flat fee of $30 plus $0.20 per mile. Write the equation for the total cost \(C\) based on the number of miles \(m\).
- **Equation**: \(C = 0.20m + 30\)
- **y-Intercept**: When \(m = 0\) (no miles driven), \(C = 30\). Thus, the y-intercept is $30.
5. **Practice Questions**:
- What does the y-intercept represent in the context of a population growth model where \(y\) is the population and \(x\) is time (in years)?
- Given the equation \(y = 3x + 5\), identify the y-intercept and interpret its meaning.
### Conclusion
The y-intercept can provide crucial insights into a problem’s initial conditions and is essential for interpreting data in various fields like finance, biology, and physics. Understanding how to identify and interpret the y-intercept allows for better analysis of real-world situations.