Graphs, Functions, And SystemsWriting An Equation And Drawing Its Graph To Model A Real-world Situation:A Construction Crew Is Lengthening A Road That Originally Measured 9 Miles. The Crew Is Adding One Mile To The Road Each Day.Let $L$ Be The

Graphs, Functions, and Systems: Modeling Real-World Situations

Introduction

In mathematics, graphs, functions, and systems are essential tools for modeling real-world situations. A graph is a visual representation of a function, while a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). Systems, on the other hand, refer to a collection of variables and their relationships. In this article, we will explore how to write an equation and draw its graph to model a real-world situation.

A Real-World Situation: Lengthening a Road

Let's consider a real-world situation where a construction crew is lengthening a road that originally measured 9 miles. The crew is adding one mile to the road each day. We can model this situation using a linear function.

Writing the Equation

To write the equation, we need to identify the variables and their relationships. Let $L$ be the length of the road after $d$ days. We know that the road originally measured 9 miles, and the crew is adding one mile to the road each day. Therefore, the length of the road after $d$ days can be represented by the equation:

$$L = 9 + d$$

This equation states that the length of the road is equal to the original length (9 miles) plus the number of days the crew has been working.

Drawing the Graph

To draw the graph, we need to plot the equation on a coordinate plane. The x-axis represents the number of days, and the y-axis represents the length of the road. We can plot a few points on the graph to get an idea of its shape.

Days (d)	Length (L)
0	9
1	10
2	11
3	12
4	13

As we can see, the graph is a straight line with a positive slope. This makes sense, since the length of the road is increasing by one mile each day.

Interpreting the Graph

The graph represents the relationship between the number of days the crew has been working and the length of the road. We can use the graph to answer questions such as:

• How long will it take to lengthen the road by 5 miles?
• What will be the length of the road after 10 days?
• How many miles will the road be lengthened each day?

To answer these questions, we can use the graph to estimate the values of $L$ and $d$.

Systems of Equations

In some cases, we may need to model a situation using a system of equations. A system of equations is a collection of two or more equations that involve the same variables. We can use systems of equations to model situations where there are multiple variables and their relationships.

Example: A Company's Revenue and Expenses

Let's consider a company that has a revenue of $100,000 and expenses of $50,000. The company's revenue is increasing by 10% each year, while its expenses are increasing by 5% each year. We can model this situation using a system of equations.

Let $R$ be the company's revenue and $E$ be its expenses. We can write the following equations:

$$R = 100,000(1 + 0.10)^t$$

$$E = 50,000(1 + 0.05)^t$$

where $t$ is the number of years.

We can solve this system of equations to find the values of $R$ and $E$ for a given value of $t$.

Conclusion

In conclusion, graphs, functions, and systems are essential tools for modeling real-world situations. By writing an equation and drawing its graph, we can model a situation and answer questions about its behavior. Systems of equations can be used to model situations where there are multiple variables and their relationships. By using these tools, we can gain a deeper understanding of the world around us and make informed decisions.

References

• [1] "Graphs and Functions" by Paul Dawkins
• [2] "Systems of Equations" by Khan Academy
• [3] "Mathematics for Real-World Situations" by James Stewart

Further Reading

• "Graph Theory" by Reinhard Diestel
• "Functions and Graphs" by Michael Sullivan
• "Systems of Linear Equations" by David C. Lay

Glossary

• Graph: A visual representation of a function.
• Function: A relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
• System: A collection of variables and their relationships.
• Equation: A statement that two expressions are equal.
• Variable: A quantity that can take on different values.
• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values for a function.
  Graphs, Functions, and Systems: Q&A

Introduction

In our previous article, we explored how to write an equation and draw its graph to model a real-world situation. We also discussed systems of equations and how they can be used to model situations where there are multiple variables and their relationships. In this article, we will answer some frequently asked questions about graphs, functions, and systems.

Q: What is the difference between a graph and a function?

A: A graph is a visual representation of a function, while a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). Think of a graph as a picture of a function, while a function is the underlying mathematical relationship.

Q: How do I determine the domain and range of a function?

A: To determine the domain and range of a function, you need to consider the possible input values (domain) and the possible output values (range). For example, if a function is defined as f(x) = 1/x, the domain would be all real numbers except 0, since you cannot divide by zero. The range would be all real numbers except 0, since the function can take on any value except 0.

Q: What is the difference between a linear function and a nonlinear function?

A: A linear function is a function that can be written in the form f(x) = mx + b, where m and b are constants. A nonlinear function is a function that cannot be written in this form. Nonlinear functions can have a variety of shapes, including quadratic, cubic, and exponential.

Q: How do I graph a function?

A: To graph a function, you need to plot points on a coordinate plane. You can use a table of values to find the x and y coordinates of the points. For example, if you have a function f(x) = x^2, you can plot the points (0,0), (1,1), (2,4), and so on.

Q: What is the difference between a system of equations and a system of inequalities?

A: A system of equations is a collection of two or more equations that involve the same variables. A system of inequalities is a collection of two or more inequalities that involve the same variables. Systems of inequalities can be used to model situations where there are multiple variables and their relationships, but the relationships are not necessarily equalities.

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to find the values of the variables that satisfy all the equations. You can use a variety of methods, including substitution, elimination, and graphing.

Q: What is the difference between a dependent variable and an independent variable?

A: A dependent variable is a variable that depends on the value of another variable. An independent variable is a variable that is not dependent on the value of another variable. For example, in the equation y = 2x, x is the independent variable and y is the dependent variable.

Q: How do I determine the order of operations in a mathematical expression?

A: To determine the order of operations in a mathematical expression, you need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Conclusion

In conclusion, graphs, functions, and systems are essential tools for modeling real-world situations. By understanding the concepts and techniques discussed in this article, you can gain a deeper understanding of the world around you and make informed decisions.

References

• [1] "Graphs and Functions" by Paul Dawkins
• [2] "Systems of Equations" by Khan Academy
• [3] "Mathematics for Real-World Situations" by James Stewart

Further Reading

• "Graph Theory" by Reinhard Diestel
• "Functions and Graphs" by Michael Sullivan
• "Systems of Linear Equations" by David C. Lay

Glossary

• Graph: A visual representation of a function.
• Function: A relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
• System: A collection of variables and their relationships.
• Equation: A statement that two expressions are equal.
• Variable: A quantity that can take on different values.
• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values for a function.
• Dependent variable: A variable that depends on the value of another variable.
• Independent variable: A variable that is not dependent on the value of another variable.