📏Honors Pre-Calculus Unit 2 – Linear Functions

Key Concepts

• Linear functions represent a constant rate of change between two variables, typically denoted as $y = mx + b$
• The slope ($m$) of a linear function determines the steepness and direction of the line
  • A positive slope indicates an increasing function, while a negative slope indicates a decreasing function
• The y-intercept ($b$) is the point where the line crosses the y-axis, representing the value of $y$ when $x = 0$
• Linear equations can be represented in various forms, including slope-intercept, point-slope, and standard form
• Parallel lines have the same slope but different y-intercepts, while perpendicular lines have slopes that are negative reciprocals of each other
• Linear functions can model real-world situations, such as cost analysis, distance-time relationships, and population growth

Graphing Linear Functions

• To graph a linear function, plot at least two points on the coordinate plane and connect them with a straight line
• The slope-intercept form ($y = mx + b$) provides a convenient way to graph a line by identifying the slope ($m$) and y-intercept ($b$)
• The x-intercept is the point where the line crosses the x-axis, representing the value of $x$ when $y = 0$
• Horizontal lines have a slope of zero and are represented by the equation $y = b$, where $b$ is the y-intercept
• Vertical lines have an undefined slope and are represented by the equation $x = a$, where $a$ is the x-intercept
• To find the x-intercept of a line, set $y = 0$ and solve for $x$
• The point of intersection between two non-parallel lines can be found by solving a system of linear equations

Slope and Intercepts

• The slope of a line can be calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are any two distinct points on the line
• The slope represents the change in $y$ over the change in $x$, or "rise over run"
  • A slope of $2$ means that for every unit increase in $x$, $y$ increases by $2$ units
• The y-intercept can be found by substituting $x = 0$ into the linear equation and solving for $y$
• The x-intercept can be found by substituting $y = 0$ into the linear equation and solving for $x$
• In the slope-intercept form ($y = mx + b$), $m$ represents the slope, and $b$ represents the y-intercept
• The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line, and $m$ is the slope

Forms of Linear Equations

• The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept
• The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line, and $m$ is the slope
  • To convert from point-slope form to slope-intercept form, simplify the equation and solve for $y$
• The standard form of a linear equation is $Ax + By = C$, where $A$, $B$, and $C$ are constants, and $A$ and $B$ are not both zero
  • To convert from standard form to slope-intercept form, solve the equation for $y$
• The intercept form of a linear equation is $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are the x-intercept and y-intercept, respectively
• To convert between different forms of linear equations, use algebraic manipulation and substitution

Parallel and Perpendicular Lines

• Parallel lines have the same slope but different y-intercepts
  • The equations of parallel lines can be written as $y = m_1x + b_1$ and $y = m_1x + b_2$, where $m_1$ is the common slope, and $b_1$ and $b_2$ are different y-intercepts
• Perpendicular lines have slopes that are negative reciprocals of each other
  • If the slope of one line is $m_1$, the slope of the perpendicular line is $m_2 = -\frac{1}{m_1}$
• To find the equation of a line parallel to a given line and passing through a specific point, use the same slope as the given line and substitute the point into the equation to find the y-intercept
• To find the equation of a line perpendicular to a given line and passing through a specific point, calculate the negative reciprocal of the given line's slope and use the point-slope form to determine the equation

Applications of Linear Functions

• Linear functions can model various real-world situations, such as:
  • Cost analysis: The total cost of producing a product based on fixed costs and variable costs per unit
  • Distance-time relationships: The distance traveled by an object moving at a constant speed over time
  • Population growth: The change in population size over time, assuming a constant growth rate
• To solve application problems, identify the relevant variables, determine the slope and y-intercept based on the given information, and create a linear equation that models the situation
• Interpret the slope and y-intercept in the context of the problem
  • For example, in a cost analysis, the slope represents the variable cost per unit, and the y-intercept represents the fixed costs
• Use the linear equation to make predictions or solve for specific values

Problem-Solving Strategies

• Read the problem carefully and identify the given information, the unknown values, and the relationships between variables
• Determine the appropriate form of the linear equation to use based on the given information
  • If the slope and y-intercept are given, use the slope-intercept form
  • If a point and the slope are given, use the point-slope form
• Substitute the known values into the chosen form of the linear equation
• Solve the equation for the unknown variable, if necessary
• Graph the linear function or interpret the results in the context of the problem
• Check your solution by substituting the values back into the original equation or comparing the results with the given information

Common Mistakes and Tips

• Remember to use the correct order of operations when simplifying linear equations
• Be careful when calculating the slope using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$; make sure to subtract the y-coordinates and x-coordinates in the correct order
• When converting from point-slope form to slope-intercept form, distribute the slope term and combine like terms before solving for $y$
• Pay attention to the signs of the slope and y-intercept when graphing linear functions
  • A positive slope indicates an increasing function, while a negative slope indicates a decreasing function
  • The y-intercept is the point where the line crosses the y-axis, not the x-axis
• When solving application problems, make sure to interpret the slope and y-intercept in the context of the problem
• Double-check your calculations and solutions to avoid errors
• Practice graphing linear functions by hand to develop a strong understanding of the relationships between the equation and the graph