Determine Exponential Growth And Decay Worksheet

6 problems

Exponential growth and decay describe changes over time. For growth, the quantity increases by a fixed percentage, modeled as $$y = a \cdot (1 + r)^t$$; the growth factor is $$1 + r$$. For decay, the quantity decreases similarly, modeled as $$y = a \cdot (1 - r)^t$$. Determine growth or decay by checking if $$r$$ is positive or negative. Use these worksheets to enhance your problem solving skills.

Algebra 2
Exponential Functions

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

• Helps students grasp how phenomena like population growth, radioactive decay, and interest rates work in real life.
• Enhances the ability to evaluate and understand exponential changes.
• Strengthens their hold close of exponential features and the way they practice extraordinary situations.
• Teaches college students to develop strategies for tackling complicated problems related to exponential growth and decay.
• Shows how exponential boom and rot are important in numerous subjects, linking science and math collectively.
• Improves abilities in studying and deciphering facts trends, that's critical for studies and facts technology.

How to Determine Exponential Growth and Decay?

• Determine if the problem describes exponential growth (increase) or decay (decrease) over time.
• Note the starting quantity $$a$$ given in the problem.
• Identify the growth rate $$r$$ for growth or the decay rate $$r$$ for decay, usually given as a percentage or decimal.
• Apply the exponential growth formula $$y = a \cdot (1 + r)^t$$ or the decay formula $$y = a \cdot (1 - r)^t$$ to solve for the quantity $$y$$ after time $$t$$.

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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