35 Linear Functions and Linear Models

Linear Functions and Linear Models is the study of functions whose rate of change is constant everywhere, formalized as functions of the form f(x) = mx + b, together with the construction and use of such functions to represent real-world situations that change at a steady, unchanging rate.

The Scope of Linear Functions

A linear function is a function whose graph is a straight line, characterized algebraically by having a constant rate of change between any two of its points — the slope computed from any pair of points on its graph is always the same value. This constant-rate property is what distinguishes a linear function from other function types, such as those whose graphs curve, where the rate of change varies depending on which two points are selected.

The Structure of a Linear Function

Every linear function can be written in the form f(x) = mx + b, where m is the slope, representing the constant rate of change, and b is the y-intercept, representing the function's output value when the input is zero. This structural form makes both defining features of a linear function directly visible: m controls the steepness and direction of the line, while b controls where the line crosses the vertical axis.

Constructing a Linear Rule

Constructing the rule for a linear function requires determining both m and b. The slope m is found using the two-point slope formula applied to any two known points on the line. Once m is known, b is found by substituting one known point's coordinates and the computed slope into the general form and solving for b, or, if one of the known points already has an x-coordinate of zero, b can be read directly as that point's y-coordinate.

Recognizing a Linear Function from a Table

A table of input-output values represents a linear function if and only if equal changes in the input always produce equal changes in the output — equivalently, if the rate of change computed between every consecutive pair of rows is identical. Checking a table for linearity means computing this rate of change across several pairs of rows and confirming it never varies; a single inconsistent rate of change is sufficient to disqualify the table from representing a linear function.

Constructing a Linear Model from Context

A linear model applies a linear function to represent a real-world scenario described in words, typically identified by phrases indicating a constant rate applied to a starting quantity, such as "starts at $50 and increases by $8 per hour." Constructing the model requires identifying the starting value as the y-intercept b and the constant rate as the slope m, then assembling the rule f(x) = mx + b using those two identified values: the scenario above produces the model f(x) = 8x + 50, where x represents hours and f(x) represents total cost.

Evaluating a Linear Model and Making Predictions

Once a linear model has been constructed, it can be evaluated at any input value using ordinary function evaluation to predict an output for a scenario not directly listed in the original data. Evaluating f(x) = 8x + 50 at x = 6 gives f(6) = 8(6) + 50 = 98, predicting that after 6 hours the total cost will be $98, provided the underlying rate genuinely remains constant across that entire range of the model's use, an assumption that should be checked against the context whenever a prediction extends meaningfully beyond the originally given data.

Distinguishing Proportional and Nonproportional Linear Functions

Every linear function is characterized by a constant rate of change, but only those with a y-intercept of exactly zero (b = 0) are also direct proportions, satisfying y = mx with no added constant. A linear function with a nonzero y-intercept, such as f(x) = 3x + 5, is linear but not proportional, since its graph does not pass through the origin and its ratio y/x is not constant across different x-values, even though its rate of change (slope) remains constant throughout. Distinguishing these two related but distinct properties — constant rate of change versus a constant ratio — prevents a common conflation between "linear" and "proportional."

Verifying a Linear Function or Model

A constructed linear function is verified by substituting each originally known point's x-coordinate into the rule and confirming the output matches that point's known y-coordinate for every point used in the construction, not merely the point used to solve for b. A linear model is further verified by checking that its predicted values remain reasonable and consistent with the real-world context described, particularly at the boundaries of the input range where the constant-rate assumption is most likely to break down.

Diagnosing Errors in Linear Functions and Models

Common errors in this area include computing the slope correctly but substituting the wrong point when solving for the y-intercept, misreading a described rate or starting value from a word problem and assigning it to the wrong parameter, assuming a table is linear after checking only one pair of rows, and confusing a linear function's slope with its y-intercept when translating between a constructed rule and its graph.