📐Algebra and Trigonometry Unit 4 – Linear Functions

Key Concepts

• Linear functions represent a constant rate of change and have the general form $y = mx + b$
• The graph of a linear function is a straight line
• Slope ($m$) measures the steepness and direction of a line
  • Calculated as the change in $y$ over the change in $x$ (rise over run)
  • Positive slope indicates an increasing function, negative slope indicates a decreasing function
• $y$-intercept ($b$) is the point where the line crosses the $y$-axis (when $x = 0$)
• $x$-intercept is the point where the line crosses the $x$-axis (when $y = 0$)
• Parallel lines have the same slope but different $y$-intercepts
• Perpendicular lines have slopes that are negative reciprocals of each other

Graphing Linear Functions

• To graph a linear function, plot at least two points and connect them with a straight line
  • One point can be the $y$-intercept (0, $b$)
  • Another point can be found by substituting an $x$-value into the equation and solving for $y$
• Alternatively, use the slope and $y$-intercept to graph the line
  • Start at the $y$-intercept and use the slope (rise over run) to plot additional points
• Identify the $x$ and $y$-intercepts of the line
• Determine the domain and range of the function
  • For a continuous linear function, the domain is all real numbers (unless otherwise specified)
  • The range depends on the slope and $y$-intercept
• Analyze the increasing/decreasing behavior and the slope of the line

Slope and Intercepts

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ for two points $(x_1, y_1)$ and $(x_2, y_2)$
• Slope-intercept form of a linear equation: $y = mx + b$
  • $m$ is the slope, and $b$ is the $y$-intercept
• Point-slope form of a linear equation: $y - y_1 = m(x - x_1)$
  • Used when given a point $(x_1, y_1)$ and the slope $m$
• Horizontal lines have a slope of zero and an equation in the form $y = b$
• Vertical lines have an undefined slope and an equation in the form $x = a$
• To find the $x$-intercept, set $y = 0$ and solve for $x$
• To find the $y$-intercept, set $x = 0$ and solve for $y$ (or identify $b$ in slope-intercept form)

Forms of Linear Equations

• Standard form: $Ax + By = C$, where $A$, $B$, and $C$ are constants, and $A$ and $B$ are not both zero
• Slope-intercept form: $y = mx + b$
  • Easily identifies slope ($m$) and $y$-intercept ($b$)
• Point-slope form: $y - y_1 = m(x - x_1)$
  • Used when a point and slope are known
• Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are the $x$ and $y$-intercepts, respectively
• Converting between forms:
  • To convert from standard form to slope-intercept form, solve for $y$
  • To convert from point-slope form to slope-intercept form, distribute the slope and simplify

Applications of Linear Functions

• Modeling real-world situations with linear functions (cost, revenue, distance, temperature)
• Interpreting slope and y-intercept in context
  • Slope represents the rate of change (cost per item, speed, etc.)
  • $y$-intercept represents the initial value or fixed cost
• Solving problems using linear models
  • Substitute known values into the linear equation and solve for the unknown variable
• Analyzing trends and making predictions based on linear models
  • Extrapolate or interpolate using the linear equation
• Identifying limitations of linear models in real-world applications
  • Some relationships may be non-linear or have constraints

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 is the point(s) of intersection that satisfy all equations
• Three types of solutions:
  • One solution (consistent and independent)
  • No solution (inconsistent)
  • Infinite solutions (consistent and dependent)
• Solving methods:
  • Graphing: plot the equations and identify the point(s) of intersection
  • Substitution: solve one equation for a variable and substitute into the other equation
  • Elimination: add or subtract equations to eliminate one variable, then solve for the other
• Applications of systems of linear equations (mixture problems, cost analysis, equilibrium)

Linear Inequalities

• Linear inequalities have the same form as linear equations but use inequality symbols (<, >, ≤, ≥)
• To graph a linear inequality:
  • Graph the corresponding equation (boundary line)
  • If the inequality is strict (< or >), use a dashed line; if inclusive (≤ or ≥), use a solid line
  • Shade the region above the line for > or ≥, or below the line for < or ≤
• Solve linear inequalities algebraically by isolating the variable on one side
  • Reverse the inequality sign when multiplying or dividing by a negative number
• Systems of linear inequalities involve two or more inequalities with the same variables
  • The solution is the region that satisfies all inequalities simultaneously
  • Shade the overlapping region that satisfies all inequalities

Common Mistakes and Tips

• Misinterpreting slope as a ratio instead of a rate of change
• Confusing the roles of $x$ and $y$ in slope calculation (rise over run, not run over rise)
• Forgetting to distribute the slope when converting from point-slope to slope-intercept form
• Incorrectly graphing inequalities (wrong shading, using the wrong line type)
• Failing to reverse the inequality sign when multiplying or dividing by a negative number
• Tips:
  • Always label your axes and clearly identify points when graphing
  • Double-check your algebra when solving equations and inequalities
  • Verify that your solution makes sense in the context of the problem
  • Practice, practice, practice! Solve a variety of problems to reinforce your understanding