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!