The image provided is an educational reference sheet (cheat sheet) comparing
Exponential Growth and
Exponential Decay. It does not contain a specific question to solve, but rather provides formulas, examples, tables, and graphs for both concepts.
Since there is no specific problem to solve, I will explain the key differences between the two concepts shown in the chart to help you understand how to use this information for your homework.
Key Differences: Exponential Growth vs. Exponential Decay
Both types of functions use the same basic equation structure:
$$y = a \cdot b^x$$
Where:
* $y$ = the final amount
* $a$ = the initial value (starting amount)
* $b$ = the growth or decay factor
* $x$ = the exponent (usually time)
#### 1. Exponential Growth
*
What happens: The quantity gets larger over time.
*
The Rule for $b$: The base $b$ must be
greater than 1 ($b > 1$).
*
Real-World Example from Chart: A population of bacteria doubling every hour.
* Equation: $y = 500 \cdot 1.05^x$
* Here, $a = 500$ (start with 500 bacteria).
* $b = 1.05$ (since $1.05 > 1$, it grows by 5% each time period).
*
Graph: The curve goes
up from left to right. As $x$ increases, $y$ increases rapidly.
#### 2. Exponential Decay
*
What happens: The quantity gets smaller over time.
*
The Rule for $b$: The base $b$ must be
between 0 and 1 ($0 < b < 1$).
*
Real-World Example from Chart: A car losing value (depreciating) each year.
* Equation: $y = 18000 \cdot 0.8^x$
* Here, $a = 18000$ (car costs $\$18,000$ new).
* $b = 0.8$ (since $0 < 0.8 < 1$, it retains 80% of its value each year, meaning it loses 20%).
*
Graph: The curve goes
down from left to right. As $x$ increases, $y$ decreases and approaches zero.
How to Use This for Homework
When you see a problem:
1.
Identify the starting amount ($a$). This is usually the number given at "time 0."
2.
Determine if it’s growing or decaying.
* If words like "increase," "double," "grow," or "appreciate" are used, it’s
Growth ($b > 1$).
* If words like "decrease," "half-life," "decay," or "depreciate" are used, it’s
Decay ($0 < b < 1$).
3.
Find the rate.
* For growth: If it grows by 5%, then $b = 1 + 0.05 = 1.05$.
* For decay: If it decays by 20%, then $b = 1 - 0.20 = 0.80$.
4.
Plug into $y = a \cdot b^x$ and solve for the unknown variable.
This chart is a great quick-reference guide. Keep it handy when doing problems involving populations, money interest, radioactive decay, or car depreciation.
Final Answer:
The image is a reference chart explaining Exponential Growth ($b > 1$) and Exponential Decay ($0 < b < 1$). No specific problem was provided to solve.
Parent Tip: Review the logic above to help your child master the concept of exponential growth and decay worksheet algebra 1.
