linear function
Understand linear functions in Grade 10 algebra. Identify functions with constant rate of change, write them in y=mx+b form, and verify linearity by checking equal differences in tables.
Key Concepts
A function with a constant rate of change is called a linear function, and its graph is a line. To check, calculate the rate of change between several pairs of points. If the rate is the same, it's linear.
Is it linear? Points: $( 4, 0), (0, 2), (2, 3)$. The rate from $( 4,0)$ to $(0,2)$ is $\frac{2 0}{0 ( 4)} = \frac{1}{2}$. The rate from $(0,2)$ to $(2,3)$ is $\frac{3 2}{2 0} = \frac{1}{2}$. Yes, it's linear! Is this linear? Points: $( 3, 0), (0, 2), (5, 4)$. The rate from $( 3,0)$ to $(0,2)$ is $\frac{2}{3}$, but from $(0,2)$ to $(5,4)$ it's $\frac{2}{5}$. Not linear!
Think of a linear function as a perfectly straight road. Its steepness, or rate of change, never varies. If the steepness changes, the road curves, and it's no longer a linear path! We can prove it's a straight line by checking if the slope is consistent between any two points.
Common Questions
What defines a linear function?
A linear function has a constant rate of change (slope) between any two points. Its graph is a straight line, and it is written as f(x) = mx + b where m is slope and b is y-intercept.
How do you verify a function is linear from a table of values?
Calculate the rate of change (Δy/Δx) between consecutive rows. If the rate is constant throughout, the function is linear. Unequal rates indicate non-linear behavior.
What is the difference between a linear function and a linear equation?
A linear equation is a single relationship (e.g., 2x + y = 5). A linear function maps inputs to outputs with constant rate of change and is written f(x) = mx + b. Functions have specific domain/range relationships.