Write and graph a linear function by examining a real-life scenario

Summary

This lesson explains how to write and graph a linear function using a real-life scenario involving taxi fares. It walks through the process of determining the cost of a taxi ride based on a flat rate and a per-mile charge, highlighting key concepts such as constant rates of change, slope-intercept form, and identifying slope and intercepts. The video demonstrates how to model the scenario with equations, tables, and graphs, reinforcing the understanding of linear relationships. It also addresses common misunderstandings and illustrates how different points on a graph can satisfy the linear equation.

Takeaways

• 😀 The cost of a taxi ride can be calculated using a linear function with a fixed rate and a flat fee.
• 🛣️ Superior Cab Company charges $0.80 per mile with a $2 flat rate, making a 4-mile ride cost $5.20.
• 📉 A linear relationship has a constant rate of change and can be represented using a table, graph, or equation.
• 🔢 Slope-intercept form (y = mx + b) helps represent linear functions, where m is the slope (rate of change) and b is the y-intercept (starting value).
• 📏 The slope represents how much the cost increases for each mile traveled, and the y-intercept is the initial charge when no miles are traveled.
• 🚖 Real-life scenarios, like taxi fares, demonstrate linear relationships where cost changes consistently with distance traveled.
• 💸 For example, a taxi company charging $1.50 per mile plus a flat fee can be modeled with an equation.
• 📐 Using known values like distance traveled and cost, you can substitute them into the equation to solve for missing variables, like the y-intercept.
• 📊 Tables can be used to calculate the total cost for different distances by plugging values into the linear equation.
• 📈 Graphs of these functions show straight lines, reinforcing the linear nature of the relationship between miles traveled and total cost.

Q & A

What is a linear relationship?

A linear relationship is a relationship between two variables with a constant rate of change. This can be observed in tables, graphs (as a straight line), and real-life scenarios.

How can you identify the slope in a linear relationship?

To find the slope in a linear relationship, you calculate the change in y-values divided by the change in x-values between two points on the line. The slope is the constant rate of change.

What does the y-intercept represent in a linear equation?

The y-intercept is the starting point of the line, where it crosses the y-axis. It represents the value of the dependent variable (y) when the independent variable (x) is 0.

What is the slope-intercept form of a linear equation?

The slope-intercept form of a linear equation is written as y = mx + b, where m is the slope and b is the y-intercept.

How does the Superior Cab Company determine the cost of a ride?

The Superior Cab Company charges a flat rate of $2 plus 80 cents for every mile traveled. This follows a linear relationship where the total cost depends on the distance traveled.

How would you calculate the cost of a four-mile cab ride with Superior Cab Company?

To calculate the cost of a four-mile cab ride, you would multiply the distance (4 miles) by 80 cents per mile and then add the flat rate of $2. This gives a total cost of $5.20.

What common mistake is made when interpreting the information in word problems?

A common mistake is incorrectly identifying key numbers, such as assuming a given distance or cost is the starting point, when it's actually an ordered pair in the linear function.

How do you model a taxi fare scenario as a linear equation?

First, identify the constant rate (slope) and any given points (like distance and cost). Use the slope-intercept form (y = mx + b) to model the scenario, where the y-intercept is determined by solving the equation with known values.

What is the purpose of creating a table when analyzing a linear scenario?

A table helps organize data by showing how the dependent variable (total cost) changes with the independent variable (distance). It allows you to calculate the cost for different distances and clearly see the linear relationship.

How can you represent a taxi fare scenario graphically?

You can graph a taxi fare scenario by plotting distance (x-axis) against cost (y-axis). The result should be a straight line that reflects the constant rate of change, with the y-intercept representing the initial flat rate.