2.5 Linear Functions and Models
Ideas in this lesson
- Recognize and simplify linear functions
- Create tables and draw linear graphs
- Interpret slopes and intercepts with scale and units
- Build models from a rate and initial value
- Recover a linear rule from numerical data
- Read a linear model from a graph
- Explore slope and intercept families
- Model grades, ramps, and unit conversions
- Compare motion and solve catch-up problems
- Build and interpret fixed-plus-variable cost models
- Prove both directions of constant-rate equivalence
Recognize and simplify linear functions · Worked example
Is \(f(x)=x \left(x + 4\right)\) linear? Explain using its simplified form and natural domain.
Answer: No; it cannot be represented by one mx+b rule on its natural domain.
The expression \(x^{2} + 4 x\) has a surviving nonlinear dependence on x.
Linear functions have constant secant slope and are defined by a first-degree or constant expression.
A nonlinear power, root, or nonconstant reciprocal does not fit mx+b here.
Linear functions have constant secant slope and are defined by a first-degree or constant expression.
A nonlinear power, root, or nonconstant reciprocal does not fit mx+b here.