Functions

Linear Function Models

Constructing a linear function to model a relationship between two quantities.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Comparing Functions

Building an Equation from a Real Situation

A phone plan bills $25 after 100 texts and $34 after 280 texts. Before reading on, think about how you'd find the plan's flat monthly fee and per-text rate from just these two bills — without being told either number directly.

Two data points from a linear relationship are enough to fully determine it, the same way two points determine a unique line. The rate of change (slope) comes from comparing the two points, and once the rate is known, either point can be used to work backward to the starting value (the y-intercept).

Worked Example — Building a Linear Model from Two Data Points

Using the phone plan bills — $25 at 100 texts, $34 at 280 texts — find the rate: (34 − 25)/(280 − 100) = 9/180 = 0.05 dollars per text. Using the point (100, 25): 25 = 0.05(100) + b, so 25 = 5 + b, giving b = 20. The model is cost = 0.05x + 20, meaning a $20 flat fee plus 5 cents per text.

Worked Example — Using a Linear Model to Predict

Using the model cost = 0.05x + 20, predict the bill for 500 texts. Substitute x = 500: cost = 0.05(500) + 20 = 25 + 20 = $45.

y = mx + b

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

Once a model is built, check it against both original data points, not just the one used to solve for the starting value — this catches an arithmetic slip before the model gets used to make any predictions.

Common Mistakes

Using the rate of change (slope) as if it were the starting value, or vice versa, when writing the final equation.
In y = mx + b, m is the rate of change and b is the starting value at x = 0 — keep track of which number represents which role before writing the final model.
Checking a built model against only the data point used to solve for b, and never verifying it against the other original point.
Substitute both original data points into the finished equation to confirm it produces the correct output for each — checking only one point can miss an error made while solving for the rate or starting value.

Key Takeaways

Two data points fully determine a linear model, the same way two points determine a unique line.
Finding the rate of change from the two points, then substituting one point back in, solves for the starting value.
A finished model can predict outputs for inputs beyond the original data, and should be checked against known data first.

Summary

Building a linear model from real data connects slope and starting value to a genuine, predictive equation. The next unit shifts from algebra to geometry, exploring how figures move and relate to each other on the coordinate plane.

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