Determine Linear Growth And Decay Worksheets [PDF]: Algebra 2 Math

How Will This Worksheet on "Determine Linear Growth and Decay" Benefit Your Student's Learning?

It enhances their ability to predict future values based on current trends, which is crucial for planning and decision-making in various fields like finance and economics.
It promotes analytical thinking by requiring students to interpret how quantities change over time, fostering a deeper understanding of mathematical concepts and their real-world applications.
Illustrates the sensible use of linear growth and rot models in regular conditions consisting of populace dynamics, monetary planning, and radioactive decay predictions.
Strengthens students' analytical capabilities by using requiring them to investigate facts and practice mathematical concepts to clear up troubles concerning increase and rot.
Improves the potential to interpret numerical records as it should be and articulate findings simply in both verbal and written formats.

Determine whether the quantity is increasing (growth) or decreasing (decay) at a constant rate over time.
Use the linear equation $ y = mx + b $, where $ m $ represents the rate of change (slope). A positive $ m $ indicates growth, while a negative $ m $ indicates decay.
Consider the starting value $ b $ (`y`-intercept) and how the quantity changes over time-based on the rate $ m $.

Solved Example

Q. How does

g(x) = 10^x

change over the interval from

x = 1

x = 3

\newline

Choices:

\newline

g(x)

increases by a factor of

20

\newline

g(x)

increases by a factor of

100

\newline

g(x)

decreases by

20\%

\newline

g(x)

decreases by a factor of

10

Solution:

Evaluate $g(x)$ at lower bound: Evaluate $g(x)$ at the lower bound of the interval. $\newline$ We need to find the value of $g(x)$ when $x = 1$ . $\newline$ Calculate $g(1) = 10^1$ .
Evaluate $g(x)$ at upper bound: Evaluate $g(x)$ at the upper bound of the interval. $\newline$ We need to find the value of $g(x)$ when $x = 3$ . $\newline$ Calculate $g(3) = 10^3$ .
Determine direction of change: Determine the direction of change in $g(x)$ over the interval. $\newline$ Compare $g(1)$ and $g(3)$ to see if $g(x)$ increases or decreases. $\newline$ Since $10^3$ is greater than $10^1$ , $g(x)$ increases from $x = 1$ to $x = 3$ .
Calculate growth factor: Calculate the factor by which $g(x)$ increases over the interval. $\newline$ Divide $g(3)$ by $g(1)$ to find the growth factor. $\newline$ Calculate the growth factor as: $\newline$ $\frac{10^3}{10^1} = 10^{3-1} = 10^2 = 100$
Match growth factor with choices: Match the calculated growth factor with the given choices. $\newline$ The growth factor is $100$ , so $g(x)$ increases by a factor of $100$ from $x = 1$ to $x = 3$ .