# Determine Linear Growth And Decay Worksheet

## 6 problems

To determine linear growth and decay, observe how a quantity changes steadily over time. In growth, the quantity increases by a constant rate per unit of time, depicted as $$y = mx + b$$, where $$m$$ represents the rate of increase. In contrast, decay involves a steady decrease at a constant rate. Use a worksheet and look at examples to understand how growth and decay are different in real-life situations.

Algebra 2
Exponential Functions

## 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.

## How to Determine Linear Growth and Decay?

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

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