Linear Function
Master linear functions written as f(x) = mx + b, where m is the constant rate of change and b is the starting value, with real-world applications in Grade 9 Algebra.
Key Concepts
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 $$f(x) = mx + b$$ 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.
Common Questions
What is a linear function?
A linear function is a function whose graph forms a straight line, written as f(x) = mx + b. The variable m represents the constant rate of change (slope), and b represents the starting value or y-intercept where the line crosses the y-axis.
How is a linear function different from a linear equation?
A linear equation like y = mx + b defines a relationship between x and y. A linear function uses function notation f(x) = mx + b, emphasizing that each input x produces exactly one output. Every linear function can be written as an equation, but function notation clarifies the input-output relationship.
What does the slope represent in a real-world linear function?
In a real-world context, the slope m represents the constant rate of change — for example, dollars earned per hour, miles traveled per hour, or items produced per minute. The y-intercept b represents the initial or starting value before any change occurs.