Linear Function
This Grade 8 math skill from Pengi Math (Grade 8) introduces the definition and properties of a linear function. Students learn that a linear function is one where the rate of change between any two points is constant, and its graph is always a straight line. The general form is y = mx + b, where m is the slope and b is the y-intercept.
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 $$y = mx + b$$ 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 $D = 50t$. A phone plan costs 20 dollars a month plus 5 cents for each text message. The monthly cost $C$ for $x$ messages is $C = 0.05x + 20$. For the function $y = 3x + 2$, 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 with a constant rate of change. Its graph is a straight line, and it can be written in the form y = mx + b.
How do you know if a function is linear?
A function is linear if the difference in y-values divided by the difference in x-values is always the same (constant slope), and its graph is a straight line.
What is the equation form of a linear function?
A linear function is written as y = mx + b, where m is the slope (rate of change) and b is the y-intercept (starting value).
Can a linear function have a slope of zero?
Yes. A slope of zero means the function is constant—the y-value never changes regardless of x, giving a horizontal line.
Where is the linear function definition taught in Grade 8?
The concept of a linear function is a core topic in the Grade 8 Pengi Math textbook under functions and linear relationships.