MTH120 College Algebra Chapter 4.2

 4.2 Modeling with Linear Functions:

Modeling with linear functions is a fundamental concept in mathematics and real-world applications. Linear functions are used to represent and describe relationships between two variables, and they have the form:

$�(�)=��+�$

Where:

• $�(�)$ is the output (dependent variable).
• $�$ is the input (independent variable).
• $�$ is the slope of the line, representing the rate of change or the relationship between the variables.
• $�$ is the y-intercept, indicating the value of $�(�)$ when $�$ is zero.

Here are the key steps for modeling with linear functions:

1. Identify the Variables:

• Determine which two variables are involved in the relationship you want to model. One variable will be the independent variable ($�$), and the other will be the dependent variable ($�(�)$).

2. Collect Data:

• If you have data related to the variables, gather and organize it. Data points should consist of pairs of values for $�$ and $�(�)$.

3. Find the Slope ($�$):

• Calculate the slope ($�$) of the linear function using the formula: $�=\frac{�({�}_2)-�({�}_1)}{{�}_2-{�}_1}$
• Select two data points (${�}_1,�({�}_1)$ and ${�}_2,�({�}_2)$) from your dataset to calculate $�$.

4. Determine the Y-Intercept ($�$):

• To find the y-intercept ($�$), you can use one of the data points and the calculated slope: $�=�({�}_1)-�\cdot {�}_1$

5. Write the Linear Function:

• Use the calculated slope ($�$) and y-intercept ($�$) to write the linear function in the form $�(�)=��+�$.

6. Interpret the Model:

• Understand the meaning of the slope and y-intercept in the context of your problem. The slope represents the rate of change, while the y-intercept is the starting value when $�=0$.

7. Use the Model:

• Once you have the linear function, you can use it to make predictions, solve problems, or analyze the relationship between the variables.

Example: Modeling with a Linear Function Suppose you have data on the number of hours worked (independent variable) and the total earnings (dependent variable) for a part-time job:

Hours Worked	Total Earnings ($)
0	10
5	30
10	50

Step 1: Identify the variables: $�$ = Hours Worked, $�(�)$ = Total Earnings.

Step 2: Collect data.

Step 3: Find the slope: $�=\frac{50-10}{10-0}=\frac{40}{10}=4$

Step 4: Determine the y-intercept: $�=10-4\cdot 0=10$

Step 5: Write the linear function: $�(�)=4�+10$

Step 6: Interpret the model: The slope of 4 indicates that for each additional hour worked, total earnings increase by $4. The y-intercept of 10 represents the starting earnings when no hours are worked.

Step 7: Use the model for predictions and analysis.

Building linear models from verbal descriptions involves translating a real-world problem or situation into a mathematical equation in the form of a linear function. To do this, follow these steps:

1. Identify the Variables:

• Read the verbal description carefully and identify the two variables involved in the relationship. One variable will be the independent variable (usually denoted as $�$), and the other will be the dependent variable (usually denoted as $�(�)$ or $�$).

2. Determine the Relationship:

• Determine whether the relationship between the variables is linear. In a linear relationship, one variable changes at a constant rate with respect to the other.

3. Write the General Form:

• Start with the general form of a linear function: $�(�)=��+�$.
  • $�(�)$ represents the dependent variable.
  • $�$ represents the independent variable.
  • $�$ is the slope of the line, indicating the rate of change or relationship.
  • $�$ is the y-intercept, representing the value of $�(�)$ when $�$ is zero.

4. Determine the Slope ($�$):

• Based on the verbal description, determine the value of the slope ($�$). The slope represents the rate of change or the coefficient that relates the variables.

5. Determine the Y-Intercept ($�$):

• Use the verbal description to determine the y-intercept ($�$), which is the value of $�(�)$ when $�$ is zero.

6. Write the Linear Model:

• Combine the determined slope ($�$) and y-intercept ($�$) with the general form to write the linear model specific to the problem.

7. Verify the Model:

• Check that the model accurately represents the verbal description and relationships between the variables.

8. Use the Model:

• Once you have the linear model, you can use it to make predictions, solve problems, or analyze the relationship between the variables.

Example: Building a Linear Model from a Verbal Description

Verbal Description: A taxi service charges a $3.50 initial fee plus $2.25 per mile traveled. Write a linear model to represent the cost of a taxi ride based on the number of miles traveled.

Step 1: Identify the Variables:

• Independent Variable ($�$): Number of miles traveled.
• Dependent Variable ($�(�)$): Cost of the taxi ride.

Step 2: Determine the Relationship:

• The cost of the taxi ride increases linearly with the number of miles traveled.

Step 3: Write the General Form:

• $�(�)=��+�$

Step 4: Determine the Slope ($�$):

• The cost per mile is $2.25, so the slope ($�$) is 2.25.

Step 5: Determine the Y-Intercept ($�$):

• The initial fee is $3.50, so the y-intercept ($�$) is 3.50.

Step 6: Write the Linear Model:

• $�(�)=2.25�+3.50$

Step 7: Verify the Model:

• The model accurately represents the cost of a taxi ride based on the number of miles traveled.

Step 8: Use the Model:

• Use the linear model to calculate the cost of a taxi ride for any given number of miles.

Using a given intercept to build a model involves constructing a linear function when you already know the y-intercept ($�$) and need to determine the slope ($�$) based on the given information. The general form of a linear function is $�(�)=��+�$, where $�(�)$ is the dependent variable, $�$ is the independent variable, $�$ is the slope, and $�$ is the y-intercept.

Here are the steps to build a linear model using a given y-intercept:

1. Identify the Variables:

• Determine the variables involved in the relationship. Identify which variable is the independent variable ($�$) and which is the dependent variable ($�(�)$ or $�$).

2. Write the General Form:

• Start with the general form of a linear function: $�(�)=��+�$.

3. Use the Given Intercept ($�$):

• Incorporate the given y-intercept ($�$) into the equation. This represents the value of $�(�)$ when $�$ is zero.

4. Determine the Slope ($�$):

• To determine the slope ($�$), you need additional information, such as a specific data point (x, f(x)), a rate of change, or a second point on the line.

5. Write the Linear Model:

• Combine the given y-intercept ($�$) and the determined slope ($�$) with the general form to write the linear model specific to the problem.

6. Verify the Model:

• Ensure that the model accurately represents the relationship between the variables and that it aligns with the given y-intercept.

7. Use the Model:

• Once you have the linear model, you can use it for predictions, calculations, or analysis related to the problem.

It's important to note that you must have sufficient information to determine the slope ($�$) in order to build a complete linear model. The y-intercept ($�$) alone is not enough to create a unique linear equation.

If you have a specific problem or example in mind where you need to build a linear model using a given intercept, please provide the details, and I can assist you further.

Using a diagram to build a model, especially a graphical representation, can be a helpful approach for visualizing and understanding linear relationships. Here's how to use a diagram to build a linear model with examples:

1. Identify the Variables:

• Determine which variables are involved in the relationship you want to model. Identify the independent variable ($�$) and the dependent variable ($�(�)$ or $�$).

2. Create a Scatterplot:

• If you have data points related to the variables, create a scatterplot to visualize their relationship. Plot the data points on a graph with the independent variable on the x-axis and the dependent variable on the y-axis.

3. Determine the Pattern:

• Examine the scatterplot to determine if there is a linear pattern or trend in the data points. A linear pattern suggests that a linear model may be appropriate.

4. Estimate the Slope ($�$):

• Use the scatterplot to estimate the slope ($�$) of the linear model. The slope represents the rate of change between the variables. You can do this by selecting two data points and calculating the rise over run ($\mathrm{\Delta }�\mathrm{/}\mathrm{\Delta }�$).

5. Estimate the Y-Intercept ($�$):

• Determine the approximate y-intercept ($�$) based on where the line intersects the y-axis on the scatterplot. The y-intercept represents the value of the dependent variable when the independent variable is zero.

6. Write the Linear Model:

• Use the estimated slope ($�$) and y-intercept ($�$) to write the linear model in the form $�(�)=��+�$.

7. Verify the Model:

• Check that the linear model accurately represents the relationship observed in the scatterplot. Ensure that it aligns with the data points.

8. Use the Model:

• Once you have the linear model, you can use it for predictions, calculations, or analysis related to the problem.

Example: Using a Diagram to Build a Linear Model

Suppose you have data on the number of study hours ($�$) and the corresponding test scores ($�(�)$) for a group of students:

Study Hours ($�$)	Test Scores ($�(�)$)
0	60
1	65
2	70
3	75

Steps:

1. Identify the Variables:

• Independent Variable ($�$): Study Hours
• Dependent Variable ($�(�)$): Test Scores

2. Create a Scatterplot:

• Plot the data points on a graph with study hours on the x-axis and test scores on the y-axis.

3. Determine the Pattern:

• Observe the scatterplot and notice that there is a linear pattern; as study hours increase, test scores also increase.

4. Estimate the Slope ($�$):

• Select two data points, such as (1, 65) and (3, 75).
• Calculate the slope: $�=\frac{75-65}{3-1}=\frac{10}{2}=5$.

5. Estimate the Y-Intercept ($�$):

• Estimate the y-intercept as approximately 60, where the line intersects the y-axis.

6. Write the Linear Model:

• Use the estimated slope ($�$) and y-intercept ($�$) to write the linear model: $�(�)=5�+60$.

7. Verify the Model:

• Check that the linear model aligns with the scatterplot.

8. Use the Model:

• You can use this linear model to predict test scores based on the number of study hours or perform other related calculations.

Using a diagram, in this case, the scatterplot, helped visually identify the linear relationship and estimate the model's parameters (slope and y-intercept).

Modeling a set of data with linear functions involves finding a linear equation that best fits the data points in order to make predictions, analyze trends, or solve problems. Here are the steps for modeling a set of data with linear functions:

1. Collect Data:

• Gather the data that you want to model. This data should consist of pairs of values for the independent variable ($�$) and the dependent variable ($�$).

2. Create a Scatterplot:

• Plot the data points on a graph with the independent variable ($�$) on the x-axis and the dependent variable ($�$) on the y-axis. This will help you visualize the relationship.

3. Determine the Linearity:

• Examine the scatterplot to determine if there is a linear trend in the data. A linear trend means that the data points approximately form a straight line.

4. Find the Best-Fitting Line:

• Identify the line that best fits the data points. This line should minimize the overall distance between the line and the data points. Methods for finding the best-fitting line include the method of least squares or regression analysis.

5. Write the Linear Model:

• Once you have found the best-fitting line, write the linear equation in the form $�=��+�$, where:
  • $�$ is the dependent variable.
  • $�$ is the independent variable.
  • $�$ is the slope of the line.
  • $�$ is the y-intercept (the value of $�$ when $�=0$).

6. Interpret the Model:

• Understand the meaning of the slope ($�$) and y-intercept ($�$) in the context of your data. The slope represents the rate of change or the relationship between the variables, while the y-intercept is the starting value.

7. Use the Model:

• Once you have the linear model, you can use it to make predictions for values of $�$ not present in your data set, analyze trends, or solve problems related to the data.

8. Verify the Model:

• Check how well the linear model fits the data. You can do this by comparing predicted values from the model to the actual data points and assessing the residuals (the differences between predicted and actual values).

9. Interpret the Results:

• Draw conclusions based on the linear model. Analyze the implications of the slope and y-intercept in the context of your data and the problem you are addressing.

10. Communicate Findings:

• Present your findings and the linear model to others, and communicate the insights and predictions that the model provides.

Example: Modeling Data with a Linear Function

Suppose you have data on the number of years of experience ($�$) and the corresponding annual salary ($�$) for a group of employees:

Years of Experience ($�$)	Annual Salary ($�$)
0	40,000
2	45,000
4	50,000
6	55,000
8	60,000

Steps:

1. Collect Data: Gather the data on years of experience and annual salary.

2. Create a Scatterplot: Plot the data points on a graph.

3. Determine the Linearity: Observe that the data points form a linear pattern.

4. Find the Best-Fitting Line: Use regression analysis or least squares to find the best-fitting line.

5. Write the Linear Model: After analysis, the linear model may be written as $�=2,500�+40,000$.

6. Interpret the Model: The slope of 2,500 indicates that for each additional year of experience, the annual salary increases by $2,500. The y-intercept of $40,000 represents the starting salary with zero years of experience.

7. Use the Model: Use the linear model to predict salaries for different years of experience or analyze the salary trends based on experience.

8. Verify the Model: Check how closely the model matches the actual data points.

9. Interpret the Results: Draw conclusions about the salary trends and the impact of years of experience on annual salary.

10. Communicate Findings: Present the linear model and insights about salary trends to others.

Modeling data with linear functions is a powerful tool for analyzing and making predictions based on real-world data sets.

Here are a few more examples of modeling data with linear functions:

Example 1: Modeling Population Growth Suppose you have data on the population of a city ($�$) over the past 10 years ($�$):

Year ($�$)	Population ($�$)
2012	100,000
2013	105,000
2014	110,000
2015	115,000
2016	120,000
2017	125,000
2018	130,000
2019	135,000
2020	140,000
2021	145,000

Steps:

• Follow the same steps as in the previous example to create a linear model to represent the population growth over time. Your linear model might look like this:
  • $�=5,000�+100,000$
• Interpret the model: The slope of 5,000 means the city's population increased by 5,000 people per year on average. The y-intercept of 100,000 represents the population at the start of 2012.

Example 2: Modeling Sales Revenue Suppose you have data on a company's monthly sales ($�$) for the past year ($�$):

Month ($�$)	Sales Revenue ($�$)
Jan	$10,000
Feb	$11,000
Mar	$12,000
Apr	$11,500
May	$12,500
Jun	$13,000
Jul	$14,000
Aug	$14,500
Sep	$15,000
Oct	$16,000
Nov	$16,500
Dec	$17,000

Steps:

• Create a scatterplot and observe that the data points roughly follow a linear pattern.
• Use regression analysis or the method of least squares to find the best-fitting line.
• Your linear model might look like this:
  • $�=500�+10,000$
• Interpret the model: The slope of 500 means that, on average, sales revenue increased by $500 per month. The y-intercept of $10,000 represents the estimated revenue in January.

Example 3: Modeling Distance vs. Time Suppose you have data on the distance traveled by a car ($�$) over time ($�$) during a road trip:

Time (hours, $�$)	Distance (miles, $�$)
0	0
1	50
2	100
3	150
4	200
5	250
6	300
7	350
8	400
9	450
10	500

Steps:

• Create a scatterplot and observe that the data points form a linear pattern.
• Use regression analysis or least squares to find the best-fitting line.
• Your linear model might look like this:
  • $�=50�$
• Interpret the model: The slope of 50 means that the car traveled 50 miles per hour. The y-intercept is zero, indicating the starting point.

These examples illustrate how to model various real-world situations with linear functions. Linear modeling is a versatile tool for understanding and making predictions based on data.