📈College Algebra Unit 4 – Linear Functions

Key Concepts and Definitions

• Linear functions expressed in the form $y = mx + b$ where $m$ represents the slope and $b$ represents the y-intercept
• Independent variable $(x)$ and dependent variable $(y)$ in a linear function
• Domain refers to the set of all possible input values $(x)$ for a function
  • For linear functions, the domain is typically all real numbers
• Range refers to the set of all possible output values $(y)$ for a function
  • For linear functions, the range is also typically all real numbers
• Slope $(m)$ measures the steepness and direction of a line
  • Positive slope indicates an increasing line, while negative slope indicates a decreasing line
• Y-intercept $(b)$ is the point where a line crosses the y-axis $(x = 0)$
• X-intercept is the point where a line crosses the x-axis $(y = 0)$

Linear Equations and Their Graphs

• Linear equations can be graphed on a coordinate plane by plotting points and connecting them with a straight line
• To graph a linear equation, find at least two points that satisfy the equation and plot them
  • One common method is to find the y-intercept $(x = 0)$ and another point using a convenient $x$ value
• The graph of a linear function is always a straight line
• Parallel lines have the same slope but different y-intercepts
• Perpendicular lines have slopes that are negative reciprocals of each other
  • For example, if one line has a slope of $2$, its perpendicular line will have a slope of $-\frac{1}{2}$
• Horizontal lines have a slope of zero and are in the form $y = b$
• Vertical lines have an undefined slope and are in the form $x = a$

Slope and Rate of Change

• Slope $(m)$ is calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ for any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line
• Slope can also be expressed as "rise over run" or $\frac{\text{change in y}}{\text{change in x}}$
• The sign of the slope indicates the direction of the line
  • Positive slope means the line is increasing from left to right
  • Negative slope means the line is decreasing from left to right
• The absolute value of the slope indicates the steepness of the line
  • A larger absolute value means a steeper line
• Slope is a measure of the rate of change of a linear function
  • It represents how much the dependent variable $(y)$ changes for each unit change in the independent variable $(x)$

Intercepts and Special Cases

• The y-intercept $(b)$ is the point where a line crosses the y-axis $(x = 0)$
  • To find the y-intercept, substitute $x = 0$ into the linear equation and solve for $y$
• The x-intercept is the point where a line crosses the x-axis $(y = 0)$
  • To find the x-intercept, substitute $y = 0$ into the linear equation and solve for $x$
• Horizontal lines have a slope of zero and are in the form $y = b$
  • The y-intercept is equal to $b$, and there is no x-intercept
• Vertical lines have an undefined slope and are in the form $x = a$
  • There is no y-intercept, and the x-intercept is equal to $a$
• A line that passes through the origin $(0, 0)$ has a y-intercept of zero
  • The equation of such a line is $y = mx$

Applications in Real-World Scenarios

• Linear functions can model many real-world situations, such as:
  • Cost analysis (fixed costs and variable costs)
  • Distance traveled over time (constant speed)
  • Temperature conversion (Celsius to Fahrenheit)
• In a cost analysis, the slope represents the variable cost per unit, and the y-intercept represents the fixed cost
• For distance traveled over time, the slope represents the speed (rate), and the y-intercept is the starting position
• Temperature conversion from Celsius to Fahrenheit follows the linear equation $F = \frac{9}{5}C + 32$
  • The slope is $\frac{9}{5}$, and the y-intercept is $32$
• Interpreting the slope and y-intercept in context is crucial for understanding the real-world implications of a linear function

Solving Systems of Linear Equations

• A system of linear equations consists of two or more linear equations with the same variables
• The solution to a system of linear equations is the point (or points) that satisfies all equations in the system
• There are three main methods for solving systems of linear equations:
  • Graphing method: Graph the equations and find the point of intersection
  • Substitution method: Solve one equation for a variable and substitute it into the other equation
  • Elimination method: Multiply equations by constants to eliminate one variable, then solve for the remaining variable
• A system of linear equations can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines)
• Real-world applications of systems of linear equations include:
  • Balancing chemical equations
  • Finding break-even points in business
  • Solving mixture problems

Common Mistakes and How to Avoid Them

• Misinterpreting the slope as the y-intercept or vice versa
  • Remember that the slope is the coefficient of $x$, and the y-intercept is the constant term
• Incorrectly calculating the slope using the "rise over run" formula
  • Make sure to subtract the y-coordinates first (rise) and the x-coordinates second (run)
• Graphing the y-intercept at $(0, x)$ instead of $(x, 0)$
  • The y-intercept is always on the y-axis, where $x = 0$
• Forgetting to distribute the negative sign when solving equations
  • Be careful when multiplying or dividing by negative numbers
• Confusing the x-intercept with the y-intercept
  • The x-intercept is where the line crosses the x-axis $(y = 0)$, while the y-intercept is where the line crosses the y-axis $(x = 0)$
• Incorrectly applying the substitution or elimination methods when solving systems of equations
  • Double-check your algebra and make sure you are performing the same operation on both sides of the equation

Practice Problems and Study Tips

• Work through a variety of practice problems to reinforce your understanding of linear functions
  • Start with simple equations and gradually increase the difficulty
• Graph linear equations by hand to develop a visual understanding of slope, intercepts, and the relationship between the equation and its graph
• Create a summary sheet with key formulas, definitions, and examples for quick reference
• When solving word problems, identify the given information, the unknown variables, and the relationships between them
  • Translate the word problem into a linear equation or system of equations
• Double-check your work by substituting your solution back into the original equation or system of equations
• Collaborate with classmates to discuss concepts, compare solutions, and learn from each other
• Seek help from your instructor or a tutor if you are struggling with a particular concept or problem
• Review your notes, textbook, and practice problems regularly to maintain your understanding of linear functions