MTH120 College Algebra Chapter 4.1

 4.1 Linear Functions:

Linear functions are a fundamental concept in mathematics and have many practical applications. A linear function is a type of function in which the highest power of the variable (usually denoted as $�$) is 1. In other words, the general form of a linear function is:

$�(�)=��+�$

Where:

• $�(�)$ is the output or dependent variable.
• $�$ is the input or independent variable.
• $�$ is the slope of the line, which determines the steepness or slant of the line.
• $�$ is the y-intercept, which is the point where the line intersects the y-axis.

Here are some key characteristics and concepts related to linear functions:

1. Slope ($�$): The slope of a linear function represents the rate of change of the dependent variable ($�$) with respect to the independent variable ($�$). It determines how steep or shallow the line is. A positive slope indicates an increasing relationship between $�$ and $�$, while a negative slope indicates a decreasing relationship.

2. Y-Intercept ($�$): The y-intercept is the value of $�$ when $�=0$. It represents the point where the line crosses or intersects the y-axis. In the equation $�(�)=��+�$, $�$ is the y-intercept.

3. Graph: The graph of a linear function is a straight line. The slope determines the direction (upward or downward) and steepness of the line, while the y-intercept determines where the line crosses the y-axis.

4. Linear Equations: Linear functions are often associated with linear equations, which can be written in the form $��+��=�$, where $�$, $�$, and $�$ are constants. These equations can be used to represent various real-world relationships.

5. Applications: Linear functions are widely used in various fields, including physics, economics, engineering, and more. They can be used to model relationships such as distance vs. time, cost vs. quantity, and many others.

6. Domain and Range: The domain of a linear function is all real numbers ($-\mathrm{\infty }$ to $+\mathrm{\infty }$) because there are no restrictions on the input $�$. The range depends on the slope $�$: if $�$ is positive, the range is also all real numbers; if $�$ is negative, the range is limited to values below the y-intercept.

Linear functions serve as a fundamental building block for more complex mathematical models and are often the first functions introduced in algebra courses. They are easy to work with and provide valuable insights into various real-world problems.

Linear functions can be represented in various ways, including through equations, graphs, tables, and verbal descriptions. Here are the different ways to represent a linear function:

1. Equation:

   • The most common way to represent a linear function is through its equation, which is typically in the form $�(�)=��+�$, where:
     • $�(�)$ is the dependent variable (usually denoted as $�$).
     • $�$ is the independent variable.
     • $�$ is the slope of the line.
     • $�$ is the y-intercept (the point where the line intersects the y-axis).

   For example, the equation $�=2�+3$ represents a linear function with a slope of 2 and a y-intercept of 3.

2. Graph:

   • Linear functions can be graphed on a coordinate plane. The slope ($�$) determines the steepness and direction of the line, while the y-intercept ($�$) determines where the line intersects the y-axis.

   For example, if the equation is $�=2�+3$, the graph would be a straight line with a slope of 2, passing through the point (0, 3).

3. Table of Values:

• You can represent a linear function by creating a table of values. Choose a few x-values, plug them into the equation, and calculate the corresponding y-values. This table shows how the function's output changes with different inputs.

4. Verbal Description:

   • Linear functions can be described verbally to explain the relationship between the variables. For example, "y is equal to 2 times x plus 3" verbally describes the function $�=2�+3$.

These representations provide different perspectives on the same linear relationship. Depending on the context and the specific problem, one representation may be more useful than others. Linear functions are essential in mathematics and have wide applications in various fields, including science, engineering, economics, and physics.




Representing a linear function in word form involves describing the relationship between the variables using natural language. To represent a linear function in word form, you can follow these steps:

1. Identify the Variables: Determine which variables are involved in the linear relationship. Typically, there is an independent variable (e.g., "x") and a dependent variable (e.g., "y").

2. Identify the Coefficients: Identify the coefficients of the variables. In a linear function of the form $�(�)=��+�$, "m" represents the slope, and "b" represents the y-intercept.

3. Describe the Relationship:

   • Start with a sentence that introduces the relationship. For example, "The variable 'y' depends on the variable 'x' in a linear manner."
   • Describe the slope ("m") by explaining how "y" changes as "x" increases. Use phrases like "for every increase of one unit in 'x'," or "as 'x' increases by one unit."
   • Describe the y-intercept ("b") by explaining the starting value of "y" when "x" is zero. Use phrases like "when 'x' is zero," or "at the point where the line crosses the y-axis."

4. Combine the Descriptions: Put all the descriptions together in a coherent sentence or paragraph to represent the linear function in word form.

Here's an example:

Linear Function: $�=2�+3$

Word Form Representation: "The variable 'y' depends on the variable 'x' in a linear manner. For every increase of one unit in 'x,' 'y' increases by two units. When 'x' is zero, 'y' is equal to three. This linear relationship is described by the equation 'y equals two times 'x' plus three.'"

In this word form representation, we've described the slope, the y-intercept, and the overall relationship between "x" and "y" in natural language.




Representing a linear function in function notation involves using the standard mathematical notation to describe the relationship between variables. To represent a linear function in function notation, you can use the general form of a linear function, which is $�(�)=��+�$, where:

• $�(�)$ is the dependent variable (usually denoted as $�$ in other contexts).
• $�$ is the independent variable.
• $�$ is the slope of the line.
• $�$ is the y-intercept.

Here's how you can represent a linear function in function notation:

1. Choose a Descriptive Variable: Instead of $�(�)$, you can use a more descriptive variable name, which often depends on the context of the problem. For example, if the linear function represents the number of miles driven ($�$) as a function of time in hours ($�$), you can define the function as $�(�)$ where $�$ stands for distance.

2. Use the General Form: Write the function in the general form $�(�)=��+�$, where $�(�)$ is replaced by the chosen variable, and $�$ remains the independent variable. For example, if you chose $�$ as the variable:

   $�(�)=��+�$

3. Specify the Parameters: Clearly state the meaning of each parameter in the context of the problem. Describe what $�$ and $�$ represent in words. For example:

   • $�$ could represent the speed in miles per hour (the rate of change of distance with respect to time).
   • $�$ could represent the initial distance at $�=0$ (the y-intercept).

4. Include Units: When appropriate, include units for the variables and parameters in your function notation. For example, if $�$ represents distance in miles and $�$ represents time in hours, you can write:

   $�(�)\text{ (in miles)}=��\text{ (in miles per hour)}+�\text{ (in miles)}$

Here's an example:

Linear Function in Function Notation: Suppose you have a linear function that represents the number of miles driven ($�$) as a function of time in hours ($�$). The function is $�(�)=50�+10$.

In this representation:

• $�(�)$ represents the distance traveled in miles as a function of time.
• $�$ is the time in hours.
• $50$ is the rate of travel in miles per hour (the slope).
• $10$ is the initial distance at $�=0$ (the y-intercept).

By using function notation, you can clearly describe the relationship and parameters of the linear function within the context of the problem.




Representing a linear function in tabular form involves creating a table that shows how the dependent variable (often denoted as $�$) changes in response to changes in the independent variable (usually denoted as $�$). To represent a linear function in tabular form, follow these steps:

1. Identify the Linear Function: Start with the linear function you want to represent. It typically has the form $�(�)=��+�$, where:

   • $�(�)$ is the dependent variable (usually $�$).
   • $�$ is the independent variable.
   • $�$ is the slope of the line.
   • $�$ is the y-intercept.

2. Choose a Range of $�$ Values: Decide on a range of $�$ values (independent variable values) that you want to include in the table. These values should cover the range of interest for your linear relationship.

3. Calculate Corresponding $�$ Values: Use the linear function to calculate the corresponding $�$ values for each of the chosen $�$ values. Substitute each $�$ value into the function $�(�)$ and compute $�$.

4. Create the Table: Organize the data into a table with two columns: one for $�$ and one for $�$. List the $�$ values and their corresponding $�$ values in rows.

Here's an example of representing a linear function in tabular form:

Linear Function: Suppose you have the linear function $�=2�+3$.

Tabular Representation:

$�$	$�$
-2	-1
-1	1
0	3
1	5
2	7

In this table:

• The chosen range of $�$ values is from -2 to 2.
• The corresponding $�$ values are calculated by substituting each $�$ value into the function $�=2�+3$.
• The table shows how the dependent variable ($�$) changes as the independent variable ($�$) varies. It provides a clear representation of the linear relationship between $�$ and $�$.

You can expand the table by including more $�$ values if needed, depending on the specific context of your linear function and the level of detail required for your analysis.




Representing a linear function in graphical form involves creating a graph on a coordinate plane that visually represents the relationship between the dependent variable ($�$) and the independent variable ($�$). Here are the steps to represent a linear function graphically:

1. Identify the Linear Function: Start with the linear function you want to graph. It typically has the form $�(�)=��+�$, where:

   • $�(�)$ is the dependent variable (usually $�$).
   • $�$ is the independent variable.
   • $�$ is the slope of the line.
   • $�$ is the y-intercept.

2. Choose a Range of $�$ Values: Decide on a range of $�$ values (independent variable values) that you want to include on the graph. These values should cover the range of interest for your linear relationship.

3. Calculate Corresponding $�$ Values: Use the linear function to calculate the corresponding $�$ values for each of the chosen $�$ values. Substitute each $�$ value into the function $�(�)$ and compute $�$.

4. Plot Points: On a coordinate plane, label the horizontal axis as $�$ and the vertical axis as $�$. Plot points with coordinates $(�,�)$, where $�$ is from the chosen range, and $�$ is the calculated value from the linear function. Each point represents a pair of ($�$, $�$) values.

5. Connect the Points: Use a straight line to connect the plotted points. This line represents the linear function.

6. Label the Axes: Label the axes with their respective variable names (e.g., $�$ and $�$) and include a scale if necessary to show the units of measurement.

Here's an example of representing a linear function graphically:

Linear Function: Suppose you have the linear function $�=2�+3$.

Graphical Representation:

• Choose a range of $�$ values (e.g., -2 to 2).
• Calculate corresponding $�$ values using the function $�=2�+3$.
• Plot points $(-2,-1)$, $(-1,1)$, $(0,3)$, $(1,5)$, and $(2,7)$ on the coordinate plane.
• Connect the points with a straight line.
• Label the horizontal axis as $�$ and the vertical axis as $�$.
• Include a scale if needed to show units of measurement.

The resulting graph will be a straight line with a slope of 2, passing through the point (0, 3), and extending in both directions. This graph visually represents the linear relationship between $�$ and $�$ described by the function $�=2�+3$.




A linear function is a type of mathematical function that represents a linear relationship between two variables, typically denoted as $�$ and $�$. The general form of a linear function is:

$�(�)=��+�$

Where:

• $�(�)$ is the dependent variable (usually denoted as $�$).
• $�$ is the independent variable.
• $�$ is the slope of the line, which determines the steepness or slant of the line.
• $�$ is the y-intercept, which is the point where the line intersects the y-axis.

Here are some examples of linear functions:

1. Example 1: Cost Function

   • Let's say you run a business, and the cost ($�$) of producing $�$ units of a product is given by: $�(�)=3�+200$
   • In this example, $�(�)$ represents the cost, $�$ represents the number of units produced, the slope $3$ represents the cost per unit, and the y-intercept $200$ represents the fixed costs.

2. Example 2: Distance-Time Relationship

   • Suppose you're driving a car at a constant speed of 60 miles per hour. The distance ($�$) traveled as a function of time ($�$) can be expressed as: $�(�)=60�$
   • Here, $�(�)$ represents the distance traveled, $�$ represents the time in hours, and the slope $60$ represents the speed of the car.

3. Example 3: Income Function

   • Let's say your monthly income ($�$) is determined by a fixed salary of $3,000 plus an additional $500 for each hour ($ℎ$) of overtime worked: $�(ℎ)=500ℎ+3000$
   • In this case, $�(ℎ)$ represents your income, $ℎ$ represents the number of hours of overtime, the slope $500$ represents the hourly rate for overtime, and the y-intercept $3000$ represents the fixed salary.

4. Example 4: Temperature Conversion

   • The Fahrenheit ($�$) temperature can be converted to Celsius ($�$) using the linear function: $�(�)=\frac{5}{9}(�-32)$
   • Here, $�(�)$ represents the temperature in Celsius, $�$ represents the temperature in Fahrenheit, and the slope $\frac{5}{9}$ and y-intercept $32$ represent the conversion formula.

In each of these examples, the linear function describes a relationship between two variables where one variable depends on the other in a linear manner. The slope ($�$) represents the rate of change, and the y-intercept ($�$) represents the starting point or value when the independent variable is zero. Linear functions are essential in various fields, including economics, physics, engineering, and many others, for modeling and analyzing relationships between variables.




You can determine whether a linear function is increasing, decreasing, or constant by examining the sign of its slope ($�$). Here's how to do it:

1. Positive Slope ($�>0$):

   • If the slope ($�$) of a linear function is positive, the function is increasing.
   • This means that as the independent variable ($�$) increases, the dependent variable ($�$) also increases.
   • The graph of the function will be an upward-sloping line from left to right.

   Example: $�(�)=2�+3$ has a positive slope of $2$, so it's an increasing function.

2. Negative Slope ($�<0$):

   • If the slope ($�$) of a linear function is negative, the function is decreasing.
   • This means that as the independent variable ($�$) increases, the dependent variable ($�$) decreases.
   • The graph of the function will be a downward-sloping line from left to right.

   Example: $�(�)=-3�+4$ has a negative slope of $-3$, so it's a decreasing function.

3. Zero Slope ($�=0$):

   • If the slope ($�$) of a linear function is zero, the function is constant.
   • This means that the value of the dependent variable ($�$) does not change as the independent variable ($�$) changes.
   • The graph of the function will be a horizontal line.

   Example: $ℎ(�)=5$ has a slope of $0$, so it's a constant function.

In summary:

• Positive slope ($�>0$): Increasing function.
• Negative slope ($�<0$): Decreasing function.
• Zero slope ($�=0$): Constant function.

By analyzing the sign of the slope, you can quickly determine the behavior of a linear function without graphing it.




Increasing and decreasing functions describe the behavior of mathematical functions as the input variable changes. These terms are commonly used to characterize how the output of a function (dependent variable) responds to changes in the input variable (independent variable).

1. Increasing Function:

   • An increasing function is a function where, as the input variable increases, the output variable also increases.
   • In other words, if you move from left to right along the graph of the function, the values of the function's output (y-values) increase.
   • Mathematically, a function $�(�)$ is increasing on an interval $[�,�]$ if, for any ${�}_1$ and ${�}_2$ in the interval $[�,�]$ where ${�}_1<{�}_2$, it is true that $�({�}_1)\le �({�}_2)$.
   • Graphically, the graph of an increasing function rises from left to right.

   Example: $�(�)=2�+1$ is an increasing function because as $�$ increases, $�(�)$ increases as well.

2. Decreasing Function:

   • A decreasing function is a function where, as the input variable increases, the output variable decreases.
   • In other words, if you move from left to right along the graph of the function, the values of the function's output (y-values) decrease.
   • Mathematically, a function $�(�)$ is decreasing on an interval $[�,�]$ if, for any ${�}_1$ and ${�}_2$ in the interval $[�,�]$ where ${�}_1<{�}_2$, it is true that $�({�}_1)\ge �({�}_2)$.
   • Graphically, the graph of a decreasing function falls from left to right.

   Example: $�(�)=-3�+4$ is a decreasing function because as $�$ increases, $�(�)$ decreases.

It's important to note that a function can have regions of both increasing and decreasing behavior within its domain. In such cases, you would specify the intervals where the function is increasing or decreasing.

For example, if you have a function $ℎ(�)$ and determine that it is increasing on the interval $[�,�]$ and decreasing on the interval $[�,�]$, you would describe it as having both increasing and decreasing behavior within its domain.




Interpreting slope as a rate of change is a fundamental concept in mathematics and has widespread applications in various fields, including physics, economics, and engineering. The slope of a linear function represents how one variable changes concerning another variable. Here's how to interpret slope as a rate of change:

Definition: The slope ($�$) of a linear function of the form $�=��+�$ represents the rate of change of the dependent variable ($�$) concerning the independent variable ($�$). It tells us how much $�$ changes for each one-unit change in $�$.

Interpretation:

1. Positive Slope ($�>0$):

   • If the slope is positive, it means that as $�$ increases, $�$ increases. The rate of change is positive, indicating a positive relationship.
   • For example, if the slope is $3$, it means that for every one-unit increase in $�$, $�$ increases by $3$. This could represent a situation like the speed of a moving object.

2. Negative Slope ($�<0$):

   • If the slope is negative, it means that as $�$ increases, $�$ decreases. The rate of change is negative, indicating an inverse relationship.
   • For example, if the slope is $-2$, it means that for every one-unit increase in $�$, $�$ decreases by $2$. This could represent a situation like the temperature decrease over time.

3. Zero Slope ($�=0$):

   • If the slope is zero, it means that $�$ remains constant as $�$ changes. There is no rate of change.
   • For example, if the slope is $0$, it means that changing $�$ has no effect on $�$. This could represent a constant value like the initial cost of a product.

4. Unit Slope ($�=1$):

   • A slope of $1$ indicates a one-to-one relationship between $�$ and $�$. For every one-unit change in $�$, $�$ changes by exactly one unit in the same direction.
   • For example, a slope of $1$ might represent a conversion rate between different units of measurement.

5. Magnitude of the Slope:

   • The magnitude of the slope ($\mathrm{\mid }�\mathrm{\mid }$) provides information about the steepness of the relationship. A larger magnitude indicates a steeper rate of change.

Understanding slope as a rate of change allows you to analyze and interpret how two variables are related in real-world situations. It's a powerful tool for making predictions, optimizing processes, and understanding the behavior of various phenomena.

Here are some examples of interpreting slope as a rate of change in real-world scenarios:

1. Distance vs. Time:

   • Consider a car traveling at a constant speed of 60 miles per hour. The slope of the distance vs. time graph is $60$, indicating that for every one-hour increase in time, the car travels an additional 60 miles.

2. Salary Increase:

   • Suppose an employee receives a $2,000 salary increase every year. The slope of the salary vs. years graph is $2,000$, showing that for each year of employment, the salary increases by $2,000.

3. Temperature Change:

   • When you cool a pot of boiling water, the temperature decreases over time. If the temperature decreases by $5$ degrees Celsius every minute, the slope of the temperature vs. time graph is $-5$.

4. Population Growth:

   • In a city with a population of 100,000 people, if the population increases by 2% per year, the slope of the population vs. years graph is $0.02$. This means the population increases by 2,000 people for every year that passes.

5. Stock Price:

   • The stock price of a company can change daily. If a stock price increases by $5 for every hour of trading, the slope of the stock price vs. time graph is $5$ dollars per hour.

6. Rainfall Accumulation:

   • During a storm, the amount of rainfall accumulates. If the rainfall accumulates at a rate of $0.5$ inches per hour, the slope of the rainfall vs. time graph is $0.5$ inches per hour.

7. Production Rate:

   • In a manufacturing plant, the production rate of a product might be $10$ units per minute. The slope of the production vs. time graph is $10$ units per minute, indicating the rate of product manufacturing.

8. Student Grades:

   • If a student's grades improve by $2$ points for every hour spent studying, the slope of the grade vs. study time graph is $2$, indicating the rate of grade improvement per hour of study.

In each of these examples, the slope of the linear relationship between two variables represents a rate of change. It quantifies how one variable changes concerning the other variable, providing valuable information for decision-making and analysis.




Writing and interpreting an equation for a linear function involves understanding how the equation relates to the real-world scenario it represents. A linear function can be expressed in the form $�(�)=��+�$, where $�(�)$ is the dependent variable, $�$ is the independent variable, $�$ is the slope, and $�$ is the y-intercept.

Here are the steps to write and interpret the equation for a linear function:

Step 1: Identify the Variables and Parameters

• Identify the dependent variable ($�(�)$) and the independent variable ($�$) in your scenario.
• Determine the slope ($�$), which represents the rate of change, and the y-intercept ($�$), which is the value of $�(�)$ when $�=0$.

Step 2: Write the Linear Equation

• Use the identified parameters to write the equation in the form $�(�)=��+�$. Replace $�(�)$ with the appropriate variable name if needed.

Step 3: Interpret the Equation

• Interpret the equation in the context of your problem. Understand what the slope and y-intercept represent.
• The slope ($�$) represents the rate of change of the dependent variable ($�(�)$) concerning the independent variable ($�$). It tells you how much $�(�)$ changes for each one-unit change in $�$.
• The y-intercept ($�$) represents the value of $�(�)$ when $�=0$, which is often a starting point or initial value.

Step 4: Apply the Equation

• Use the equation to make predictions, analyze data, or solve specific problems related to your scenario.

Example: Writing and Interpreting a Linear Function Equation

Scenario: A taxi service charges a $2.50 base fare plus $1.75 per mile driven.

Step 1: Identify Variables and Parameters

• Dependent variable: Total fare ($�$)
• Independent variable: Number of miles driven ($�$)
• Slope ($�$): Rate per mile driven, which is $1.75
• Y-intercept ($�$): Base fare, which is $2.50

Step 2: Write the Linear Equation

• The equation for the total fare ($�$) as a function of the number of miles driven ($�$) is: $�(�)=1.75�+2.50$

Step 3: Interpret the Equation

• The slope ($1.75$) represents the rate at which the total fare increases for each additional mile driven.
• The y-intercept ($2.50$) represents the initial cost of the taxi ride when no miles have been driven.

Step 4: Apply the Equation

• You can use the equation to calculate the total fare for any given number of miles driven. For example, if a passenger travels 5 miles, the total fare would be F(5) = 1.75(5) + 2.50 = $10.25.

Writing and interpreting linear function equations allows you to model and analyze real-world situations, make predictions, and understand how variables are related in various contexts.




Modeling real-world problems with linear functions involves using linear equations to represent and analyze relationships between variables in various practical scenarios. Linear functions are particularly useful for modeling situations where there is a constant rate of change. Here's a step-by-step approach to modeling real-world problems with linear functions:

Step 1: Identify the Variables

• Identify the variables involved in the problem. Typically, you'll have one dependent variable (usually denoted as $�$) and one or more independent variables (usually denoted as $�$, $�$, or other letters).

Step 2: Understand the Situation

• Gain a clear understanding of the real-world situation or problem. What is happening, and how are the variables related? Pay attention to any constant rates of change or initial values.

Step 3: Write the Linear Equation

• Based on your understanding of the problem, write a linear equation that relates the dependent variable ($�$) to the independent variable(s) ($�$, $�$, etc.). The equation should have the form $�=��+�$, where $�$ is the slope and $�$ is the y-intercept.
• Determine the values of $�$ and $�$ based on the problem's information.

Step 4: Interpret the Equation

• Interpret the equation in the context of the problem. What does the slope represent, and what does the y-intercept represent?
• The slope ($�$) represents the rate of change of $�$ concerning $�$. It tells you how much $�$ changes for each one-unit change in $�$.
• The y-intercept ($�$) represents the value of $�$ when $�$ is zero, which often corresponds to an initial value or starting point.

Step 5: Apply the Equation

• Use the equation to make predictions, answer specific questions, or analyze the problem further.
• Solve for $�$ or $�$ when specific values are given, or use the equation to calculate values for different scenarios.

Step 6: Check the Model

• Ensure that the linear model accurately represents the real-world situation. Check if it aligns with your observations and the problem's context.

Here are some common real-world scenarios that can be modeled with linear functions:

1. Distance vs. Time: Modeling the distance traveled by a moving object over time.
2. Cost vs. Quantity: Modeling the cost of a product or service based on the quantity purchased.
3. Temperature vs. Time: Modeling how temperature changes over time.
4. Income vs. Experience: Modeling how income increases with years of experience in a job.
5. Population vs. Year: Modeling population growth or decline over time.

By applying these steps, you can create accurate linear models for a wide range of real-world problems, making it easier to analyze data, make predictions, and solve practical challenges.




Graphing linear functions is a fundamental skill in mathematics and is used to visually represent the relationship between two variables. Linear functions have equations of the form $�=��+�$, where $�$ is the slope of the line, and $�$ is the y-intercept (the point where the line crosses the y-axis).

Here's how to graph a linear function step by step, along with examples:

Step 1: Determine the Slope and Y-Intercept

• Look at the equation of the linear function to identify the values of $�$ and $�$.
• $�$ is the slope, which tells you how steep the line is. If $�>0$, the line slopes upward to the right; if $�<0$, it slopes downward.
• $�$ is the y-intercept, which is the point where the line crosses the y-axis (when $�=0$).

Step 2: Plot the Y-Intercept

• Begin by plotting the point $(0,�)$ on the graph, where $�$ is the y-intercept you found in step 1.

Step 3: Use the Slope to Find More Points

• Use the slope $�$ to find one or more additional points on the line.
• The slope $�$ represents the change in $�$ for each change of $�$ by 1 unit. For example, if $�=2$, for each unit increase in $�$, $�$ increases by 2 units, and for each unit decrease in $�$, $�$ decreases by 2 units.

Step 4: Plot the Additional Points

• Plot the additional points you found in step 3 on the graph.

Step 5: Draw the Line

• Connect the plotted points with a straight line. This line represents the graph of the linear function.

Now, let's go through a couple of examples:

Example 1: Graphing $�=2�+3$

1. Determine the slope and y-intercept:

   • $�=2$ (slope)
   • $�=3$ (y-intercept)

2. Plot the y-intercept:

   • The y-intercept is $(0,3)$.

3. Use the slope to find more points:

   • For each unit increase in $�$, $�$ increases by 2 units. So, if $�=1$, then $�=5$ ($1$ unit right and $2$ units up from the y-intercept).
   • You can also choose $�=-1$, which gives $�=1$ ($1$ unit left and $2$ units down from the y-intercept).

4. Plot the additional points:

   • $(1,5)$ and $(-1,1)$.

5. Draw the line:

   • Connect the three points to form a straight line.

Example 2: Graphing $�=-0.5�+2$

1. Determine the slope and y-intercept:

   • $�=-0.5$ (slope)
   • $�=2$ (y-intercept)

2. Plot the y-intercept:

   • The y-intercept is $(0,2)$.

3. Use the slope to find more points:

   • For each unit increase in $�$, $�$ decreases by 0.5 units. So, if $�=1$, then $�=1.5$ ($1$ unit right and $0.5$ units down from the y-intercept).
   • You can also choose $�=-1$, which gives $�=2.5$ ($1$ unit left and $0.5$ units up from the y-intercept).

4. Plot the additional points:

   • $(1,1.5)$ and $(-1,2.5)$.

5. Draw the line:

   • Connect the three points to form a straight line.

These examples demonstrate how to graph linear functions by finding the slope and y-intercept and using that information to plot points and draw the line.

Graphing a function by plotting points involves selecting values of the independent variable, evaluating the function to find corresponding values of the dependent variable, and then plotting these points on a coordinate plane to create the function's graph. Here are the steps to graph a function by plotting points:

Step 1: Understand the Function

• Begin by understanding the given function and its domain (the range of valid input values) and range (the set of possible output values).

Step 2: Choose Values of the Independent Variable

• Select a set of values for the independent variable (typically denoted as $�$) that cover the desired range of the function.
• It's a good practice to choose both positive and negative values, as well as zero, to get a sense of how the function behaves in different regions.

Step 3: Evaluate the Function

• For each chosen value of $�$, calculate the corresponding value of the dependent variable ($�$) by using the function's equation.
• Substitute each $�$ value into the function and compute $�$. This gives you ordered pairs $(�,�)$ representing points on the graph.

Step 4: Plot the Points

• On a coordinate plane, plot each ordered pair $(�,�)$ as a point.
• Label the points if necessary.

Step 5: Connect the Points

• After plotting the points, connect them with a smooth curve or line that represents the function.
• Ensure that the curve accurately reflects the behavior of the function between the plotted points.

Step 6: Label the Axes and Title

• Label the x-axis and y-axis with appropriate units and scales.
• Add a title or description to the graph, especially if it represents a real-world situation.

Here's an example to illustrate the process:

Example: Graphing the Function $�(�)=2{�}^2-3�+1$

Step 1: Understand the Function

• The function $�(�)=2{�}^2-3�+1$ is a quadratic function.

Step 2: Choose Values of the Independent Variable

• Select a range of $�$ values, such as $�=-2$, $�=-1$, $�=0$, $�=1$, and $�=2$ to cover a portion of the function's domain.

Step 3: Evaluate the Function

• Calculate the corresponding $�$ values by substituting each chosen $�$ value into the function:
  • $�(-2)=2(-2{)}^2-3(-2)+1=9$
  • $�(-1)=2(-1{)}^2-3(-1)+1=6$
  • $�(0)=2(0{)}^2-3(0)+1=1$
  • $�(1)=2(1{)}^2-3(1)+1=0$
  • $�(2)=2(2{)}^2-3(2)+1=5$

Step 4: Plot the Points

• Plot the ordered pairs: $(-2,9)$, $(-1,6)$, $(0,1)$, $(1,0)$, and $(2,5)$ on the coordinate plane.

Step 5: Connect the Points

• Connect the points with a smooth curve to create the graph of the function.

Step 6: Label the Axes and Title

• Label the x-axis as "x" and the y-axis as "f(x)."
• Add a title to the graph, such as "Graph of $�(�)=2{�}^2-3�+1$."

By following these steps, you can graph functions by plotting points and visually represent their behavior on a coordinate plane.




Graphing a function using its y-intercept and slope is a straightforward method for drawing a straight-line graph when you have the equation in slope-intercept form, which is $�=��+�$, where:

• $�$ is the dependent variable (usually represented on the vertical axis).
• $�$ is the independent variable (usually represented on the horizontal axis).
• $�$ is the slope of the line.
• $�$ is the y-intercept, the point where the line crosses the y-axis ($�=0$).

Here are the steps to graph a function using the y-intercept and slope:

Step 1: Identify the Slope and Y-Intercept

• Look at the equation of the function to identify the values of $�$ (slope) and $�$ (y-intercept).

Step 2: Plot the Y-Intercept

• Start by plotting the point $(0,�)$ on the graph. This point represents the y-intercept.

Step 3: Use the Slope to Find Additional Points

• The slope $�$ tells you the rate at which $�$ changes concerning $�$.
• If the slope is a fraction $�=\frac{�}{�}$, where $�$ and $�$ are integers, this means that for each one-unit increase in $�$, $�$ changes by $\frac{�}{�}$ units.
• To find more points on the line, use the slope to calculate the change in $�$ for specific changes in $�$. For example, if $�=\frac{2}{3}$, for each one-unit increase in $�$, $�$ increases by $\frac{2}{3}$ units.

Step 4: Plot the Additional Points

• Plot the additional points you found in step 3 on the graph.

Step 5: Connect the Points

• Connect all the plotted points with a straight line. This line represents the graph of the function.

Here's an example to illustrate the process:

Example: Graphing the Function $�=\frac{2}{3}�+1$

Step 1: Identify the Slope and Y-Intercept

• $�=\frac{2}{3}$ (slope)
• $�=1$ (y-intercept)

Step 2: Plot the Y-Intercept

• The y-intercept is $(0,1)$.

Step 3: Use the Slope to Find Additional Points

• Since $�=\frac{2}{3}$, for each one-unit increase in $�$, $�$ increases by $\frac{2}{3}$ units.
• To find more points, you can choose $�=3$, which gives $�=\frac{2}{3}(3)+1=3$.

Step 4: Plot the Additional Points

• Plot the point $(3,3)$.

Step 5: Connect the Points

• Connect the two points, $(0,1)$ and $(3,3)$, with a straight line.

The resulting graph is a straight line that represents the function $�=\frac{2}{3}�+1$. This method allows you to quickly graph linear functions when you know the y-intercept and slope.




Graphing a function using transformations involves adjusting a base function's graph by applying transformations such as translations, reflections, stretches, and compressions. Transformations allow you to modify the shape, position, and size of the graph. The base function is typically a simple, known function like $�=�$, $�=\sin (�)$, or $�={�}^{�}$. Here are the steps to graph a function using transformations:

Step 1: Start with a Base Function

• Begin with a known base function that represents the untransformed version of your function.

Step 2: Identify the Transformations

• Determine which transformations need to be applied to the base function. These transformations include:
  • Translations (shifting the graph horizontally or vertically).
  • Reflections (flipping the graph over the x-axis, y-axis, or other lines).
  • Stretches or compressions (changing the size of the graph vertically or horizontally).

Step 3: Apply the Transformations

• For each identified transformation, apply it to the base function's equation. Adjust the coefficients and constants accordingly.
• If you're shifting the graph, add or subtract values inside the function to shift it vertically or horizontally.
• If you're reflecting the graph, apply a negative sign to one or both variables.
• If you're stretching or compressing the graph, adjust the coefficients.

Step 4: Graph the Transformed Function

• Plot key points and use the transformed equation to create the graph.
• It's often helpful to create a table of values by selecting different $�$ values, evaluating the transformed function to find the corresponding $�$ values, and then plotting the points on the graph.

Step 5: Label the Axes and Title

• Label the x-axis and y-axis with appropriate units and scales.
• Add a title to the graph, especially if it represents a real-world situation.

Here's an example to illustrate the process:

Example: Graphing the Transformed Function $�=-2\sin (2�+�)+3$

Step 1: Start with a Base Function

• The base function is $�=\sin (�)$.

Step 2: Identify the Transformations

• The transformations are as follows:
  • Vertical stretch by a factor of 2 (multiplied by -2).
  • Horizontal compression by a factor of 2 (inside the sine function).
  • Horizontal translation to the left by $�$ radians (inside the sine function).
  • Vertical translation upward by 3 units (added 3).

Step 3: Apply the Transformations

• Apply each transformation to the base function's equation:
  • Vertical stretch: $�=-2\sin (�)$
  • Horizontal compression: $�=-2\sin (2�)$
  • Horizontal translation: $�=-2\sin (2�+�)$
  • Vertical translation: $�=-2\sin (2�+�)+3$

Step 4: Graph the Transformed Function

• Plot key points or use a table of values to create the graph of $�=-2\sin (2�+�)+3$.

Step 5: Label the Axes and Title

• Label the x-axis and y-axis as needed.
• Add a title to the graph, such as "Graph of $�=-2\sin (2�+�)+3$."

By following these steps, you can graph functions using transformations to achieve the desired modifications to the base function's graph.

To write the equation for a function from the graph of a line, you need to identify key information from the graph, such as the slope and y-intercept. The equation of a line is typically in the form $�=��+�$, where:

• $�$ represents the dependent variable.
• $�$ represents the independent variable.
• $�$ is the slope of the line.
• $�$ is the y-intercept, the point where the line crosses the y-axis ($�=0$).

Here are the steps to write the equation for a function from the graph of a line:

Step 1: Identify the Slope (m)

• Determine the slope of the line by looking at the steepness or inclination of the line on the graph.
• The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Choose two points that are easily recognizable.

Step 2: Identify the Y-Intercept (b)

• Locate the point where the line crosses the y-axis (the vertical axis). This point represents the y-intercept.
• Note the y-coordinate of this point.

Step 3: Write the Equation

• Once you have determined the slope ($�$) and the y-intercept ($�$), you can write the equation for the line in the form $�=��+�$.
• Substitute the values of $�$ and $�$ into the equation.

Step 4: Simplify (if Necessary)

• Simplify the equation if possible. For example, if the slope is a fraction, you can reduce it.

Step 5: Add Context (if Applicable)

• If the line represents a real-world situation, add context to the equation by describing what the variables $�$ and $�$ represent in that context.

Here's an example:

Example: Writing the Equation from the Graph

Suppose you have the graph of a line, and it appears to have a slope of 2 (meaning it rises 2 units for every 1 unit it runs) and a y-intercept of 3 (crossing the y-axis at the point (0, 3)).

Step 1: Identify the Slope ($�$)

• The slope is 2.

Step 2: Identify the Y-Intercept ($�$)

• The y-intercept is 3.

Step 3: Write the Equation

• The equation for the line is $�=2�+3$.

Step 4: Simplify (if Necessary)

• No simplification is needed in this case.

Step 5: Add Context (if Applicable)

• If this line represents, for example, the cost ($�$) of $�$ items, you can add context by saying, "The cost ($�$) of $�$ items is given by the equation $�=2�+3$."

That's how you can write the equation for a function from the graph of a line by identifying the slope and y-intercept and using them in the standard linear equation form.




Finding the x-intercept of a line involves determining the point(s) at which the line crosses the x-axis. The x-intercept(s) represent the values of $�$ for which $�=0$. To find the x-intercept, follow these steps:

Step 1: Set $�$ to 0

• Since the x-intercept is the point where $�=0$, set $�$ to 0 in the equation of the line.

Step 2: Solve for $�$

• Solve the equation for $�$ to find the x-coordinate(s) of the x-intercept(s). Depending on the equation, there may be one or more x-intercepts.

Here's an example to illustrate the process:

Example: Finding the x-Intercept of the Line $�=2�-4$

Step 1: Set $�$ to 0

• $0=2�-4$

Step 2: Solve for $�$

• Add 4 to both sides of the equation: $4=2�$
• Divide both sides by 2 to isolate $�$: $�=2$

So, the x-intercept of the line $�=2�-4$ is the point (2, 0), where the line crosses the x-axis.

In some cases, a line may not intersect the x-axis at all, which means it has no x-intercepts. In other cases, it may intersect the x-axis at multiple points, resulting in multiple x-intercepts.

Keep in mind that finding the x-intercept is useful for understanding the behavior of the line and its relationship to the x-axis. It represents the values of $�$ where the line crosses or touches the x-axis, making $�$ equal to 0 at those points.




Horizontal and vertical lines are fundamental types of lines in mathematics, each with distinct characteristics. Here's how to describe them:

Horizontal Line:

• A horizontal line is a straight line that runs parallel to the x-axis on a Cartesian plane.
• It has a constant y-coordinate throughout its entire length.
• The equation of a horizontal line is in the form $�=�$, where $�$ is a constant representing the y-coordinate.
• The slope of a horizontal line is zero (0).
• It never intersects the y-axis, and its y-intercept is at the point where it crosses the y-axis, which is $(0,�)$.
• All points on a horizontal line have the same y-coordinate, making it easy to determine.
• Examples of horizontal lines include lines like $�=3$, $�=-5$, and $�=0$.

Vertical Line:

• A vertical line is a straight line that runs parallel to the y-axis on a Cartesian plane.
• It has a constant x-coordinate throughout its entire length.
• The equation of a vertical line is in the form $�=�$, where $�$ is a constant representing the x-coordinate.
• The slope of a vertical line is undefined because it has no change in the x-coordinate between any two points on the line.
• It never intersects the x-axis, and its x-intercept is at the point where it crosses the x-axis, which is $(�,0)$.
• All points on a vertical line have the same x-coordinate, making it easy to determine.
• Examples of vertical lines include lines like $�=4$, $�=-2$, and $�=0$.

In summary, horizontal lines have a constant y-coordinate and are described by equations of the form $�=�$, while vertical lines have a constant x-coordinate and are described by equations of the form $�=�$. Understanding these characteristics makes it easy to identify and describe horizontal and vertical lines on a coordinate plane.




To determine whether two lines are parallel or perpendicular, you need to examine their slopes. The slope of a line indicates how steep it is and its direction. Here are the key concepts:

Slope of a Line (m):

• The slope of a line represents the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.
• Mathematically, if $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ are two points on the line, the slope $�$ is calculated as: $�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$.
• For a horizontal line, the slope is always 0 because there is no vertical change.
• For a vertical line, the slope is undefined because there is no horizontal change.

Now, let's determine whether two lines are parallel or perpendicular:

Parallel Lines:

• Two lines are parallel if they have the same slope (i.e., ${�}_1={�}_2$).
• Parallel lines have the same steepness and will never intersect, no matter how far they are extended.

Perpendicular Lines:

• Two lines are perpendicular if the product of their slopes is equal to -1 (i.e., ${�}_1\cdot {�}_2=-1$).
• Perpendicular lines intersect at a right angle (90 degrees) and have slopes that are negative reciprocals of each other.

Here are some examples:

Example 1: Determining Parallel Lines

• Line 1: $�=2�+3$
• Line 2: $�=2�-1$

Both lines have the same slope (${�}_1=2$ and ${�}_2=2$), so they are parallel.

Example 2: Determining Perpendicular Lines

• Line 1: $�=-3�+4$
• Line 2: $�=\frac{1}{3}�+2$

To determine if they are perpendicular, find the slopes: ${�}_1=-3$ and ${�}_2=\frac{1}{3}$. Then, check if ${�}_1\cdot {�}_2=-1$. In this case, $(-3)\cdot \left(\frac{1}{3}\right)=-1$, so the lines are perpendicular.

By analyzing the slopes of the lines, you can determine whether they are parallel or perpendicular. Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.




To write the equation of a line that is parallel or perpendicular to a given line, you need to follow specific steps based on the properties of parallel and perpendicular lines. Let's explore both cases:

1. Writing the Equation of a Line Parallel to a Given Line:

Given the equation of a line in the form $�=��+�$, where $�$ is the slope and $�$ is the y-intercept, and you want to find the equation of a line parallel to it:

Step 1: Find the slope (${�}_1$) of the given line.

Step 2: Use the same slope (${�}_1$) for the parallel line.

Step 3: Optionally, choose a point $({�}_0,{�}_0)$ through which the new parallel line passes. This can be any point, but it's often convenient to use integers.

Step 4: Write the equation of the parallel line using the slope-intercept form $�=��+�$, where $�$ is the same as the slope of the given line, and $�$ is found by substituting the chosen point's coordinates $({�}_0,{�}_0)$ into the equation.

2. Writing the Equation of a Line Perpendicular to a Given Line:

Given the equation of a line in the form $�=��+�$, where $�$ is the slope and $�$ is the y-intercept, and you want to find the equation of a line perpendicular to it:

Step 1: Find the slope (${�}_1$) of the given line.

Step 2: Determine the negative reciprocal of the slope of the given line. The negative reciprocal is $-\frac{1}{{�}_1}$.

Step 3: Optionally, choose a point $({�}_0,{�}_0)$ through which the new perpendicular line passes.

Step 4: Write the equation of the perpendicular line using the slope-intercept form $�=��+�$, where $�$ is the negative reciprocal of the slope of the given line, and $�$ is found by substituting the chosen point's coordinates $({�}_0,{�}_0)$ into the equation.

Here are examples for both cases:

Example 1: Writing the Equation of a Line Parallel to $�=2�+3$

Given the line $�=2�+3$, the slope is ${�}_1=2$.

Suppose you want a line parallel to it with a point $(1,1)$.

Step 1: ${�}_1=2$ (slope of the given line).

Step 2: Use the same slope, $�=2$.

Step 3: Given point $(1,1)$.

Step 4: Write the equation of the parallel line: $�=2�-1$.

Example 2: Writing the Equation of a Line Perpendicular to $�=-3�+4$

Given the line $�=-3�+4$, the slope is ${�}_1=-3$.

Suppose you want a line perpendicular to it with a point $(2,-1)$.

Step 1: ${�}_1=-3$ (slope of the given line).

Step 2: The negative reciprocal of -3 is $-\frac{1}{-3}=\frac{1}{3}$.

Step 3: Given point $(2,-1)$.

Step 4: Write the equation of the perpendicular line: $�=\frac{1}{3}�-\frac{7}{3}$.

By following these steps, you can find the equations of lines that are parallel or perpendicular to a given line with a known equation

To write the equation of a line that is perpendicular to a given line, follow these steps:

Step 1: Find the slope (${�}_1$) of the given line.

Step 2: Determine the negative reciprocal of the slope of the given line. The negative reciprocal is $-\frac{1}{{�}_1}$.

Step 3: Choose a point $({�}_0,{�}_0)$ through which the new perpendicular line passes. This point can be any point, but it's often convenient to use integers.

Step 4: Write the equation of the perpendicular line using the point-slope form or the slope-intercept form.

Let's go through an example:

Example: Writing the Equation of a Line Perpendicular to $�=2�+3$ through the Point $(1,1)$

Given the line $�=2�+3$, the slope is ${�}_1=2$.

Step 1: ${�}_1=2$ (slope of the given line).

Step 2: The negative reciprocal of 2 is $-\frac{1}{2}$.

Step 3: Given point $(1,1)$.

Step 4: Write the equation of the perpendicular line.

Using Point-Slope Form: The point-slope form of a line is $�-{�}_0=�(�-{�}_0)$. Substituting the values, we have:

$�-1=-\frac{1}{2}(�-1)$

Now, you can simplify and rewrite it in the slope-intercept form if needed:

$�-1=-\frac{1}{2}�+\frac{1}{2}$

$�=-\frac{1}{2}�+\frac{3}{2}$

So, the equation of the line that is perpendicular to $�=2�+3$ and passes through the point $(1,1)$ is $�=-\frac{1}{2}�+\frac{3}{2}$.

This equation represents a line with a slope of $-\frac{1}{2}$ and passes through the point $(1,1)$, making it perpendicular to the given line.