Curriculum placement
Algebra 2. This describes the skill, not a learner estimate.
Linear growth against exponential growth
Justify linear versus exponential growth and compare eventual polynomial growth
Problem types
- Linear or exponential
- When exponential overtakes linear
- Choose the table of values that shows exponential decay
- Find the first whole input where an exponential exceeds a linear function
Builds on
- Zeros and factors of a polynomial
- Building an exponential model
- Comparing functions given in different forms · Earlier idea
Comparing functions in different forms extends to asking which kind of growth eventually exceeds the other.
- Solving exponential and logarithmic equations · Useful idea
Finding when an exponential quantity overtakes a linear one means solving an exponential equation with logarithms.
- Properties of logarithms · Useful idea
Log properties turn the equation that compares two exponentials into a linear equation in the unknown time.
- Logarithms as inverses of exponentials · Useful idea
The time at which an exponential overtakes another quantity is found by taking a logarithm.
- Comparing functions · Earlier idea
Comparing rates of change of two functions extends to comparing linear and exponential growth over long times.
Leads to
- Solving a linear and a nonlinear equation together · See it applied
Where a linear and an exponential model are equal is a solution of a system with a curve.
- Sums of a finite geometric sequence · See it applied
The total of an exponentially growing quantity over equal steps is a geometric series.
- Checking a model against evidence · See it applied
Choosing between linear and exponential models by how the data grow is part of checking whether a model fits.
Related skills
- End behavior of a polynomial · Related idea
A polynomial grows slower than any exponential in the long run, which growth comparison shows.
- Rules for an arithmetic sequence · Related idea
An arithmetic sequence adds a constant each step and a geometric sequence multiplies. Comparing them shows linear against exponential growth.
- Representing a proportional relationship · Related idea
A proportional relationship adds the same amount per unit, while exponential growth multiplies by the same factor, so growth-comparison contrasts them.
- Quadratic, rational, and exponential constraints · Shared foundations
Open the interactive map to explore lessons and skill connections.