Linear Data Has Constant First Differences
Linear data is identified by constant first differences: when x-values are evenly spaced, if the change in consecutive y-values (Δy) is the same for every pair, the data follows a linear pattern. In Grade 11 enVision Algebra 1 (Chapter 8: Quadratic Functions), students learn to compute first differences and use a constant result as the indicator of linearity. This constant difference equals the slope of the linear function that models the data, directly linking the table analysis to the equation.
Key Concepts
When data follows a linear pattern , the first differences between consecutive $y$ values are constant.
For data points with evenly spaced $x$ values, if $\Delta y = y {i+1} y i$ is the same for all consecutive pairs, then the data can be modeled by a linear function.
Common Questions
What are first differences in a data table?
First differences are the differences between consecutive y-values: Δy = y_{i+1} − y_i. They show how much the output changes from one x-step to the next.
What does it mean if first differences are constant?
Constant first differences indicate the data follows a linear pattern and can be modeled by a linear function.
How do constant first differences relate to the slope?
The constant first difference equals the slope m of the linear function that models the data.
What type of data produces non-constant first differences?
Quadratic data has constant second differences (not first differences). Exponential data has first differences that grow proportionally rather than staying constant.
How do you calculate first differences from a table?
Subtract each y-value from the next: compute y₂ − y₁, y₃ − y₂, y₄ − y₃, etc. If all results are equal, the data is linear.
Can first differences be negative?
Yes. A constant negative first difference indicates a linear function with a negative slope — y decreases by the same amount with each x-step.