Lesson 4.1: Linear Functions

New Concept

Linear functions describe relationships with a constant rate of change. We'll explore how to write their equations, like $f(x) = mx + b$, interpret slope and intercepts, and graph them to model and solve real-world problems.

What’s next

Next, you'll work through interactive examples on finding slope and writing linear equations. Then, test your skills with practice cards and challenge problems.

Property

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

where $b$ is the initial or starting value of the function (when input, $x = 0$), and $m$ is the constant rate of change, or slope of the function. The $y$-intercept is at $(0, b)$.

Examples

• A car travels at a constant speed of 50 miles per hour. Its distance $D$ from a starting point after $t$ hours can be modeled by $D(t) = 50t$.
• A phone plan costs 20 dollars a month plus 5 cents for each text message. The monthly cost $C$ for $x$ messages is $C(x) = 0.05x + 20$.
• For the function $f(x) = 3x + 2$, the value when $x=4$ is $f(4) = 3(4) + 2 = 14$. The point $(4, 14)$ is on the line.

Explanation

Think of a linear function as a rule for anything that changes at a steady rate. The 'm' is the rate of change (how steep the line is), and 'b' is the starting point on the vertical axis before any change happens.

Property

The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function.

Examples

• The function $f(x) = 2x + 5$ is increasing because the slope $m=2$ is positive.
• The function $g(x) = -3x + 1$ is decreasing because the slope $m=-3$ is negative.
• The function $h(x) = 9$ is constant because the slope $m=0$. Its graph is a horizontal line.

Explanation

The sign of the slope tells you the direction of the line. A positive slope means the line goes uphill from left to right. A negative slope means it goes downhill. A zero slope means the line is perfectly flat.

Property

The slope, or rate of change, of a function $m$ can be calculated according to the following:

where $x_1$ and $x_2$ are input values, $y_1$ and $y_2$ are output values. To interpret the slope, use the units of the output and input, such as 'dollars per hour' or 'feet per second'.

Examples

• Given the points $(2, 1)$ and $(6, 9)$, the slope is $m = \frac{9-1}{6-2} = \frac{8}{4} = 2$.
• A town's population grew from 5,200 to 6,000 in 4 years. The rate of change is $\frac{6000 - 5200}{4 \text{ years}} = \frac{800}{4} = 200$ people per year.
• If a linear function has points $(1, -1)$ and $(4, 5)$, its slope is $m = \frac{5 - (-1)}{4 - 1} = \frac{6}{3} = 2$. The function is increasing.

Explanation

Slope measures the steepness of a line. It's the 'rise over run,' telling you how much the vertical value changes for every one unit of horizontal change. A bigger slope value means a steeper line.

Property

To write an equation for a linear function, you need to find the slope $(m)$ and the $y$-intercept $(b)$.

1. Identify two points on the line.
2. Use the two points to calculate the slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$.
3. Use the slope and one point $(x_1, y_1)$ in the point-slope form, $y - y_1 = m(x - x_1)$, and solve for $y$. Alternatively, substitute $m$ and a point into $y = mx+b$ and solve for $b$.

Examples

• A line has a slope of 4 and passes through $(2, 5)$. Using $y = mx + b$, we get $5 = 4(2) + b$, so $5 = 8 + b$, and $b = -3$. The equation is $y = 4x - 3$.
• A line passes through $(1, 2)$ and $(4, 11)$. The slope is $m = \frac{11-2}{4-1} = \frac{9}{3} = 3$. Using the point $(1,2)$, we have $y-2 = 3(x-1)$, which simplifies to $y = 3x - 1$.
• A gym charges a 50 dollars sign-up fee and 25 dollars per month. The cost function is $C(x) = 25x + 50$, where $x$ is the number of months.

Explanation

To define a specific line, you need to know its direction (slope) and one point it passes through. Once you have these two pieces of information, you can create a unique formula for that line.

Property

The $x$-intercept of the function is value of $x$ when $f(x) = 0$. It is the point where the graph crosses the $x$-axis. It can be solved by the equation $0 = mx + b$. Not all linear functions have $x$-intercepts; a horizontal line like $y=c$ (for $c \neq 0$) does not cross the $x$-axis.

Examples

• To find the x-intercept of $f(x) = 2x - 10$, set $0 = 2x - 10$. Solving gives $10 = 2x$, so $x = 5$. The x-intercept is $(5, 0)$.
• The function $f(x) = -\frac{1}{3}x + 2$ has an x-intercept where $0 = -\frac{1}{3}x + 2$. This gives $\frac{1}{3}x = 2$, so $x=6$. The intercept is $(6, 0)$.
• The horizontal line $f(x)=4$ never crosses the x-axis, so it has no x-intercept.

Explanation

The x-intercept is where the line hits the horizontal axis. At this specific point, the output of the function is zero. To find it, you just set the function's formula equal to zero and solve for x.

Property

Lines can be horizontal or vertical.
A horizontal line is a line defined by an equation in the form $f(x) = b$. It has a slope of 0.
A vertical line is a line defined by an equation in the form $x = a$. Its slope is undefined. A vertical line is not a function.

Examples

• The equation of a horizontal line passing through the point $(5, -2)$ is $y = -2$.
• The equation of a vertical line passing through the point $(3, 8)$ is $x = 3$.
• The function $f(x) = 10$ represents a horizontal line with a slope of 0. The line $x = -1$ is a vertical line with an undefined slope.

Explanation

A horizontal line is perfectly flat because its y-value is always the same, resulting in a zero slope. A vertical line is straight up and down because its x-value is constant. Its slope is undefined.

Property

Two lines are parallel lines if they do not intersect. The slopes of the lines are the same ($m_1 = m_2$) and the $y$-intercepts are different.
Two lines are perpendicular lines if they intersect to form a right angle. The product of their slopes is $-1$ ($m_1 m_2 = -1$), meaning their slopes are negative reciprocals of each other ($m_2 = -\frac{1}{m_1}$).

Examples

• The lines $f(x) = 4x + 1$ and $g(x) = 4x - 5$ are parallel because both have a slope of $4$.
• The lines $f(x) = 2x + 3$ and $g(x) = -\frac{1}{2}x + 8$ are perpendicular because their slopes, $2$ and $-\frac{1}{2}$, multiply to $-1$.
• The line parallel to $y = -x + 2$ that passes through $(1, 5)$ is $y = -x + 6$.

Explanation

Parallel lines have the same steepness, so they never meet. Perpendicular lines intersect at a perfect 90-degree angle. Their slopes have a special relationship: one is the flipped, negated version of the other, like $3$ and $-\frac{1}{3}$.