Linear functions are the building blocks of algebra. They're everywhere in math and real life, from calculating costs to modeling relationships between variables. Understanding their behavior and how to represent them is crucial for problem-solving.

Linear functions have a constant rate of change, represented by the slope. This makes them predictable and easy to work with. By mastering linear functions, you'll have a solid foundation for tackling more complex mathematical concepts and real-world applications.

Linear Functions

Representation of linear functions

• Linear function equation $y = mx + b$ represents a straight line on a coordinate plane
  • $m$ represents the slope, which is the rate of change or steepness of the line (rise over run)
  • $b$ represents the y-intercept, which is the value of $y$ when $x = 0$ (where the line crosses the y-axis)
• Graphing linear functions involves using the slope and y-intercept to plot points and draw the line
  • Slope determines the steepness and direction of the line
    • Positive slope: line rises from left to right (uphill)
    • Negative slope: line falls from left to right (downhill)
    • Zero slope: horizontal line (flat)
    • Undefined slope: vertical line (straight up and down)
• Real-world scenarios can be modeled using linear functions
  • Cost per unit (slope) and fixed costs (y-intercept) for a business
  • Distance traveled over time with constant speed (slope) and starting position (y-intercept)
  • Temperature change with altitude, where temperature decreases at a constant rate (slope) from a starting temperature (y-intercept)

Behavior of linear functions

• Rate of change, or slope, represents the change in $y$ per unit change in $x$
  • Calculated using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line
  • Indicates how quickly the function increases or decreases
• Y-intercept is the point where the line crosses the y-axis $(0, b)$
  • Represents the initial value or starting point in a real-world context
  • For a business, the y-intercept might represent the fixed costs before any products are sold

Slope in various contexts

• Slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ calculates the slope between any two points on a line
• Parallel lines have the same slope but different y-intercepts
  • Equations: $y = m_1x + b_1$ and $y = m_1x + b_2$, where $m_1$ is the same for both lines
  • Example: multiple roads with the same incline but different starting elevations
• Perpendicular lines have slopes that are negative reciprocals of each other
  • If $m_1$ is the slope of one line, the slope of the perpendicular line is $m_2 = -\frac{1}{m_1}$
  • Example: the slope of a roof (rise over run) and the slope of the gutter (run over rise)

Forms of linear equations

• Point-slope form: $y - y_1 = m(x - x_1)$
  • Used when given a point $(x_1, y_1)$ on the line and the slope $m$
  • Helpful for finding the equation of a line passing through a specific point
• Slope-intercept form: $y = mx + b$
  • Used when given the slope $m$ and y-intercept $b$
  • Can be derived from point-slope form by simplifying the equation
  • Most common form for representing linear functions
• Function notation: $f(x) = mx + b$ is another way to express a linear function, where $f(x)$ represents the output (dependent variable) for a given input $x$ (independent variable)

Linear functions in real-world modeling

• Identify the independent variable (input) and dependent variable (output)
• Determine the slope and y-intercept based on the given information
  • Slope represents the rate of change, such as cost per unit or speed
  • Y-intercept represents the initial value or starting point, such as fixed cost or initial distance
• Interpret the meaning of the slope and y-intercept in the context of the problem
• Use the linear function to make predictions and solve problems
  • Example: predicting the total cost for a specific number of units sold
  • Example: determining the time required to travel a certain distance at a constant speed

Domain, Range, and Variation

• Domain: the set of all possible input values (x-values) for a linear function
  • For most linear functions, the domain is all real numbers
  • In real-world contexts, the domain may be restricted (e.g., time cannot be negative)
• Range: the set of all possible output values (y-values) for a linear function
  • For most linear functions, the range is all real numbers
  • In real-world contexts, the range may be restricted (e.g., temperature cannot exceed a certain value)
• Direct variation: a special case of linear functions where $y = kx$ (no y-intercept)
  • The constant $k$ is called the constant of variation
  • Example: distance traveled is directly proportional to time when speed is constant
• Linear regression: a statistical method used to find the best-fitting linear function for a set of data points
  • Used to model relationships between variables and make predictions based on observed data

Applying Linear Functions

Representation of linear functions

• A car rental company charges a fixed fee of $50 plus $0.25 per mile driven
  • Independent variable: miles driven $(x)$
  • Dependent variable: total cost $(y)$
  • Slope: $0.25 (cost per mile)
  • Y-intercept: $50 (fixed fee)
  • Equation: $y = 0.25x + 50$
• Graphing the car rental cost function
  • Plot the y-intercept $(0, 50)$
  • Use the slope to plot additional points, such as $(100, 75)$ for 100 miles driven
  • Connect the points to create the line representing the cost function

Behavior of linear functions

• Comparing the cost of two different phone plans
  • Plan A: $30 fixed fee + $0.10 per minute
    • Slope: $0.10 (cost per minute)
    • Y-intercept: $30 (fixed fee)
    • Equation: $y = 0.10x + 30$
  • Plan B: $0.20 per minute with no fixed fee
    • Slope: $0.20 (cost per minute)
    • Y-intercept: $0 (no fixed fee)
    • Equation: $y = 0.20x$
  • Analyze which plan is more cost-effective based on usage
    • Plan A is better for low usage (less than 150 minutes)
    • Plan B is better for high usage (more than 150 minutes)

Slope in various contexts

• Finding the slope of a roof
  • Rise: vertical distance from the bottom to the top of the roof
  • Run: horizontal distance from the bottom to the top of the roof
  • Slope = $\frac{rise}{run}$
  • Example: a roof with a rise of 4 feet and a run of 12 feet has a slope of $\frac{4}{12} = \frac{1}{3}$
• Designing a ramp with a specific slope to meet accessibility requirements
  • Parallel ramps: same slope, different starting points
    • Example: two ramps with a slope of $\frac{1}{12}$ but different heights
  • Perpendicular ramps: negative reciprocal slopes
    • Example: a ramp with a slope of $\frac{1}{12}$ is perpendicular to a ramp with a slope of $-12$

Forms of linear equations

• Finding the equation of a line passing through $(3, 5)$ with a slope of $2$
  1. Use the point-slope form: $y - 5 = 2(x - 3)$
  2. Simplify the equation: $y - 5 = 2x - 6$
  3. Solve for $y$: $y = 2x - 1$
• The resulting equation in slope-intercept form is $y = 2x - 1$

Linear functions in real-world modeling

• Modeling the relationship between temperature and altitude
  • Temperature decreases by $6.5°C$ per $1000$ meters increase in altitude
  • Slope: $-0.0065$ (temperature change per meter)
  • Y-intercept: $T_0$ (temperature at sea level)
  • Equation: $T = -0.0065h + T_0$, where $T$ is temperature, $h$ is altitude, and $T_0$ is temperature at sea level
  • Example: if the temperature at sea level is $20°C$, the equation would be $T = -0.0065h + 20$
• Interpreting the break-even point in a business context
  • Revenue: $R = px$, where $p$ is price per unit and $x$ is the number of units sold
  • Cost: $C = mx + b$, where $m$ is the variable cost per unit, $x$ is the number of units produced, and $b$ is the fixed cost
  • Break-even point: where revenue equals cost $(R = C)$
    1. Set up the equation: $px = mx + b$
    2. Solve for $x$: $x = \frac{b}{p - m}$
    3. Interpret the break-even point as the minimum number of units that must be sold to cover all costs
  • Example: if p = 10, m = 6, and b = $1000, the break-even point is $x = \frac{1000}{10 - 6} = 250$ units