2.1 Linear Functions

Linear functions are the foundation for nearly everything you'll encounter in pre-calculus and beyond. They describe relationships where one quantity changes at a constant rate relative to another. Because that rate of change never varies, linear functions produce straight lines on a graph and are the most predictable type of function you'll work with.

Mastering how to write, graph, and interpret linear functions now will pay off when you move into quadratics, polynomials, and eventually calculus, where "rate of change" becomes the central idea.

Linear Functions

Representation of linear functions

The standard equation for a linear function is $y = mx + b$, which produces a straight line on the coordinate plane.

• $m$ is the slope: the rate of change, or how much $y$ changes for every 1-unit increase in $x$. Think of it as rise over run.
• $b$ is the y-intercept: the value of $y$ when $x = 0$. On the graph, it's where the line crosses the y-axis.

To graph a linear function, start by plotting the y-intercept $(0, b)$, then use the slope to find additional points. For example, if $m = 3$, move 1 unit right and 3 units up from the y-intercept to get your next point. Connect the points with a straight line.

The slope controls the line's direction:

• Positive slope: line rises from left to right
• Negative slope: line falls from left to right
• Zero slope: horizontal line (no change in $y$)
• Undefined slope: vertical line (not actually a function, since it fails the vertical line test)

Real-world scenarios modeled by linear functions include:

• A business where the slope is cost per unit and the y-intercept is the fixed cost
• Travel at constant speed, where slope is speed and the y-intercept is starting position
• Temperature decreasing at a steady rate with altitude, where slope is the rate of decrease and the y-intercept is the starting temperature

Behavior of linear functions

The rate of change (slope) tells you how quickly the output changes relative to the input. You can calculate it from any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

A key property of linear functions is that this ratio is the same no matter which two points you pick. If you calculate different slopes from different pairs of points in a data set, the relationship isn't perfectly linear.

The y-intercept $(0, b)$ represents the starting value before the independent variable has any effect. In a business model, for instance, the y-intercept is the fixed cost you pay before selling a single product.

Slope in various contexts

The slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ works between any two distinct points on a line.

Parallel lines share the same slope but have different y-intercepts. Their equations look like $y = mx + b_1$ and $y = mx + b_2$, where $b_1 \neq b_2$. Because the slopes match, the lines never intersect. Think of two roads with the same incline but at different elevations.

Perpendicular lines have slopes that are negative reciprocals of each other. If one line has slope $m_1$, the perpendicular line has slope:

$m_2 = -\frac{1}{m_1}$

For example, if a line has slope $\frac{2}{3}$, any line perpendicular to it has slope $-\frac{3}{2}$. Multiplying the two slopes always gives $-1$, which is a quick way to check perpendicularity.

Forms of linear equations

You'll use different forms depending on what information you're given.

• Point-slope form: $y - y_1 = m(x - x_1)$
  • Best when you know a point $(x_1, y_1)$ and the slope $m$
  • Especially useful for writing equations quickly from a graph or word problem
• Slope-intercept form: $y = mx + b$
  • Best when you know (or want to identify) the slope and y-intercept directly
  • The most common form for graphing and interpreting linear functions
• Function notation: $f(x) = mx + b$
  • Replaces $y$ with $f(x)$ to emphasize that the output depends on the input $x$
  • $f(3) = 2(3) - 1 = 5$ means "when the input is 3, the output is 5"

You can always convert between forms. Point-slope form converts to slope-intercept form by distributing and solving for $y$.

Linear functions in real-world modeling

When modeling a scenario with a linear function, follow these steps:

1. Identify the variables. Determine which quantity is the independent variable (input, $x$) and which is the dependent variable (output, $y$).
2. Find the slope. This is the rate of change: cost per unit, speed, degrees per meter, etc.
3. Find the y-intercept. This is the initial value: fixed cost, starting position, base temperature, etc.
4. Write the equation in the form $y = mx + b$.
5. Interpret in context. State what the slope and y-intercept mean in the real-world situation.
6. Use the equation to predict. Plug in values of $x$ to find corresponding values of $y$, or vice versa.

Domain, Range, and Variation

Domain is the set of all possible input values ($x$-values). For a pure linear function, the domain is all real numbers. In applied problems, the domain is often restricted: you can't drive negative miles or work negative hours.

Range is the set of all possible output values ($y$-values). Like domain, it's all real numbers for a general linear function but may be limited in context (e.g., cost can't be negative).

Direct variation is a special linear function of the form $y = kx$, where the y-intercept is zero. The line passes through the origin, and $k$ is called the constant of variation. Distance equals rate times time ($d = rt$) is a classic example: when time is zero, distance is zero.

Linear regression is a statistical method for finding the best-fitting line through a set of data points that don't fall perfectly on a line. Your calculator or software minimizes the overall distance between the data points and the line, producing an equation you can use for predictions.

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: $y = 0.25x + 50$

To graph this function:

1. Plot the y-intercept at $(0, 50)$
2. Use the slope: for every 100 miles, cost increases by $25, giving the point $(100, 75)$
3. Connect the points with a straight line

At 200 miles, the total cost is $y = 0.25(200) + 50 = 100$, so $100.

Behavior of linear functions

Comparing two phone plans shows how slope and y-intercept affect cost:

• Plan A: $30 fixed fee + $0.10 per minute → $y = 0.10x + 30$
• Plan B: $0.20 per minute, no fixed fee → $y = 0.20x$

To find where the plans cost the same, set the equations equal:

$0.10x + 30 = 0.20x$ $30 = 0.10x$ $x = 300 \text{ minutes}$

Below 300 minutes, Plan B is cheaper (lower starting cost wins). Above 300 minutes, Plan A is cheaper (lower per-minute rate wins). At exactly 300 minutes, both cost $60.

Slope in various contexts

Finding the slope of a roof:

• Rise: vertical height (say, 4 feet)
• Run: horizontal distance (say, 12 feet)
• Slope: $\frac{4}{12} = \frac{1}{3}$

Accessibility ramps use the same idea. The ADA requires a maximum slope of $\frac{1}{12}$, meaning 1 inch of rise for every 12 inches of run.

• Parallel ramps have the same slope ($\frac{1}{12}$) but start at different heights
• Perpendicular ramps have negative reciprocal slopes: a ramp with slope $\frac{1}{12}$ is perpendicular to one with slope $-12$

Forms of linear equations

Finding the equation of a line through $(3, 5)$ with slope $2$:

1. Start with point-slope form: $y - 5 = 2(x - 3)$

2. Distribute: $y - 5 = 2x - 6$

3. Add 5 to both sides: $y = 2x - 1$

The slope-intercept form $y = 2x - 1$ tells you the line crosses the y-axis at $(0, -1)$ and rises 2 units for every 1 unit to the right.

Linear functions in real-world modeling

Temperature and altitude: Temperature drops by about $6.5°C$ for every 1000 meters of elevation gain. This is called the lapse rate.

• Slope: $-0.0065$ °C per meter (negative because temperature decreases)
• Y-intercept: $T_0$ (temperature at sea level)
• Equation: $T = -0.0065h + T_0$

If sea-level temperature is $20°C$, then at $3000$ meters: $T = -0.0065(3000) + 20 = 0.5°C$.

Break-even analysis: A company wants to know how many units it must sell to cover costs.

• Revenue: $R = px$ (price per unit times units sold)
• Cost: $C = mx + b$ (variable cost per unit times units, plus fixed costs)

To find the break-even point:

1. Set revenue equal to cost: $px = mx + b$

2. Solve for $x$: $x = \frac{b}{p - m}$

3. This gives the minimum number of units to sell before turning a profit

If the price per unit is $10, variable cost is $6, and fixed costs are $1000:

$x = \frac{1000}{10 - 6} = 250 \text{ units}$

The company must sell at least 250 units to break even. Every unit beyond 250 generates $4 of profit.