Desmos Graphing Calculator
Type a linear equation like y = 5x + 30 to see and explore its graph.
Open DesmosLearn the Concepts
1 · Functions
A function takes each input to exactly one output — written y = f(x), where x is the input (independent variable) and y is the output (dependent variable). Functions can be shown four ways: in words, as a table, as a graph, or as an equation.
The domain is the set of allowed inputs; the range is the set of resulting outputs.
Learn the Concepts
2 · Linear models
A relationship is linear when the output changes by a constant amount for each unit of input — a constant rate of change (the slope). In slope-intercept form:
y = m·x + b (m = rate of change, b = initial value when x = 0)
The slope between two points is the change in output over the change in input:
m = (y₂ − y₁) ÷ (x₂ − x₁)
Graph linear models quickly with the Desmos graphing calculator linked below.
Worked Examples
See it done, step by step
Example 1 — A gym charges a $30 fee plus $5 per visit. Model the cost.
The $30 is the initial value (b); the $5 per visit is the rate of change (m).
C = 5x + 30. For 8 visits: C = 5(8) + 30 = $70.
Example 2 — Find the slope between (2, 50) and (5, 80)
m = (80 − 50) ÷ (5 − 2) = 30 ÷ 3 = 10 (the output rises 10 per unit of input).
Example 3 — A car is 100 miles away and approaches at 60 mph. Model its distance.
Distance starts at 100 and decreases 60 each hour, so the rate of change is negative.
d = 100 − 60t. After 1 hour: d = 100 − 60 = 40 miles.
Watch & Review
Resources for this module
Lecture materials
Module 13 lecture slides and any posted videos are available in your Canvas course.
Open CanvasCheck Yourself
Quick self-check
1. In y = 3x + 12, what are the rate of change and the initial value?
Rate of change = 3; initial value = 12.
2. A phone plan is $20 per month plus $0.10 per minute. Cost for 100 minutes?
C = 20 + 0.10(100) = 20 + 10 = $30.
3. What makes a model linear rather than nonlinear?
A constant rate of change — the output changes by the same amount for each step in the input.
Ready to turn it in?
When linear models make sense, complete the Module 13 homework and quiz in Canvas.
Submit in Canvas