Question

Linear functions from situations practice identify the slope

Answer 1

Identifying the slope in linear functions involves understanding the relationship between two variables and how they change together. Here are a few examples of situations that can represent linear functions. For each situation, we'll identify the slope.

### Example Situations:

1. **Paying for Gas**:
- A gas station charges $3 per gallon. If you buy \( x \) gallons of gas, your total cost \( C \) is given by the linear function \( C = 3x \).
- **Slope**: 3 (This means for every extra gallon of gas, the cost increases by $3.)

2. **Walking Distance**:
- You walk at a constant speed of 2 miles per hour. The distance \( d \) you cover in \( t \) hours can be expressed as \( d = 2t \).
- **Slope**: 2 (This means for every hour you walk, you cover 2 more miles.)

3. **Temperature Change**:
- A furnace heats a room such that for every hour it runs, the temperature increases by 5 degrees. The temperature \( T \) after \( h \) hours can be modeled as \( T = 5h + T_0 \), where \( T_0 \) is the initial temperature.
- **Slope**: 5 (This means for each hour, the temperature rises by 5 degrees.)

4. **Concert Ticket Sales**:
- A concert sells tickets for $25 each. If \( n \) tickets are sold, the total revenue \( R \) can be written as \( R = 25n \).
- **Slope**: 25 (This signifies that for each additional ticket sold, revenue increases by $25.)

5. **Monthly Subscription Fee**:
- A video streaming service charges a flat fee of $10 per month. If \( m \) represents months, the total cost \( C \) can be described as \( C = 10m \).
- **Slope**: 10 (This indicates that for each month of service, the cost increases by $10.)

### General Formula for Slope

The slope \( m \) in a linear function \( y = mx + b \) can be determined as follows:

- **Interpretation of Slope**: The slope represents the rate of change between the dependent variable (y) and the independent variable (x). It tells you how much y changes for a one-unit increase in x.

When given a situation, you can identify the slope by:

1. **Identifying Variables**: Determine which quantity is changing with respect to the other.
2. **Finding the Rate of Change**: Look at how much the dependent variable increases (or decreases) when the independent variable increases by 1 unit.

### If you have specific situations you'd like to analyze or if you need more practice problems, feel free to ask!