4.1 Linear Functions

Linear functions describe relationships that change at a constant rate. They show up constantly in real-world scenarios like calculating costs, modeling motion, or predicting trends. Understanding their forms and features gives you the foundation for nearly everything else in algebra.

Linear Functions

Forms of Linear Functions

There are three standard ways to write a linear equation, and each one highlights different information about the line.

Slope-intercept form: $y = mx + b$

• $m$ is the slope (rate of change)
• $b$ is the y-intercept (the value of $y$ when $x = 0$)
• This is the most common form and the easiest to graph from directly.

Point-slope form: $y - y_1 = m(x - x_1)$

• $m$ is the slope
• $(x_1, y_1)$ is any known point on the line
• Use this when you know the slope and one point but not the y-intercept.

Standard form: $Ax + By = C$

• $A$, $B$, and $C$ are constants, with $A$ and $B$ not both zero
• To extract useful info: the slope is $\frac{-A}{B}$ and the y-intercept is $\frac{C}{B}$, as long as $B \neq 0$
• This form is especially useful for finding intercepts quickly (set $x = 0$ or $y = 0$).

Slope as Rate of Change

Slope measures how much $y$ changes for every one-unit increase in $x$. It's calculated as:

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

• Positive slope: the function increases (line goes up left to right)
• Negative slope: the function decreases (line goes down left to right)
• Zero slope: horizontal line, meaning $y$ doesn't change at all
• Undefined slope: vertical line, meaning $x$ doesn't change (and this is not a function)

In real-world contexts, slope always has units. On a distance-time graph, slope represents speed (miles per hour, meters per second, etc.). On a cost graph, slope might represent price per item.

Direct variation is a special case where the y-intercept is zero: $y = kx$. Here $k$ is called the constant of variation, and it's the slope. The line passes through the origin.

Equations from Given Information

Different starting information calls for different strategies:

Given slope and y-intercept: Plug directly into slope-intercept form $y = mx + b$.

Given slope and a point: Use point-slope form, then simplify.

• Example: slope $m = 3$, point $(2, 5)$
• $y - 5 = 3(x - 2)$, which simplifies to $y = 3x - 1$

Given two points:

1. Calculate the slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

2. Plug $m$ and either point into point-slope form

3. Simplify to slope-intercept form

Given a table of values: Pick any two points from the table and follow the two-points method above. Check that the slope is the same between every pair of consecutive points to confirm the relationship is actually linear.

Given a graph: Read the y-intercept directly from where the line crosses the y-axis. Then pick two clear points on the line to calculate the slope.

Graphing and Key Features

To graph a linear function from slope-intercept form:

1. Plot the y-intercept at $(0, b)$
2. From that point, use the slope as "rise over run" to find a second point. For example, a slope of $\frac{2}{3}$ means go up 2 and right 3.
3. Plot the second point and draw a straight line through both.

Key features to identify on any linear graph:

• y-intercept: where the line crosses the y-axis (set $x = 0$)
• x-intercept: where the line crosses the x-axis (set $y = 0$ and solve)
• Slope: the steepness and direction of the line

Relationships Between Lines

Parallel lines have the same slope but different y-intercepts. They never intersect. If one line is $y = 2x + 3$, then $y = 2x - 7$ is parallel to it.

Perpendicular lines have slopes that are negative reciprocals of each other. Their product is $-1$:

$m_1 \cdot m_2 = -1$

So if one line has slope $2$, a perpendicular line has slope $-\frac{1}{2}$. If one line has slope $-\frac{3}{4}$, a perpendicular line has slope $\frac{4}{3}$.

Real-World Applications of Linear Functions

Word problems involving linear functions follow a consistent process:

1. Identify what the variables represent (what's the input, what's the output)
2. Pull out the slope (rate of change) and a known point or y-intercept (starting value)
3. Build the linear equation
4. Solve for the unknown
5. Check that your answer makes sense in context (no negative distances, for instance)

Linear interpolation lets you estimate values between known data points by assuming the relationship is linear between them. This is useful when you have a table of data and need a value that falls between two entries.

Representations of Linear Functions

Linear functions can be expressed in four ways, and you should be comfortable moving between all of them:

• Algebraic (equations): Slope-intercept, point-slope, or standard form. Each highlights different properties.
• Graphical: A visual line on the coordinate plane. You can quickly read intercepts, slope direction, and whether the function is increasing or decreasing.
• Tabular (tables of values): Lists of specific $(x, y)$ pairs. A constant difference in $y$-values for equal steps in $x$ confirms linearity.
• Verbal (word problems): A description of a real-world linear relationship. Your job is to translate the words into one of the other representations.

Function Notation and Domain/Range

Function notation $f(x)$ replaces $y$ and emphasizes that the output depends on the input. Writing $f(3) = 7$ means "when $x = 3$, the output is 7."

• Domain: the set of all possible input values ($x$-values). For most linear functions, the domain is all real numbers.
• Range: the set of all possible output values ($y$-values). For most linear functions, the range is also all real numbers.

The exception is a horizontal line like $f(x) = 4$, which has a domain of all real numbers but a range of just $\{4\}$.