Question 3 Carly receives $ every week for babysitting for her neighbors. She decides to save the money she receives to help pay for a camp she wants to attend next summer. She tracks how much money she saves from babysitting over the course of the year. Choose a dollar amount that you think Carly might receive each week. Explain why Carly’s savings over time is an example of linear change. Create a sequence that describes Carly’s growing savings each week over the course of 20 weeks. Create a formula for Carly’s savings over time. On a sheet of graph paper, create a graph for Carly’s savings over time. Describe at least one key feature on your graph that signals it depicts linear change. Question 4 When Nico was born, his grandparents put $ in a special savings account. The amount of money in the savings account increases by 8% every year. Nico tracks how much money is in the account over time. Choose a dollar amount that you think Nico’s grandparents might have put in the savings account when Nico was born. Explain why the money in Nico’s savings account over time is an example of exponential growth. Create a sequence that describes the amount of money in Nico’s savings account each year for 20 years. Create a formula for the amount of money in Nico’s savings account over time. On a sheet of graph paper, create a graph for the amount of money in Nico’s savings account over time. Describe at least one key feature on your graph that signals it depicts exponential growth. Question 5 Aria buys a new car for $. She learns that every year her car will depreciate in value by 15%. This means that each year her car will be worth 85% of what it was the previous year. Aria tracks how much her car is worth over time. Choose a dollar amount that you think Aria might have purchased her car for. Explain how the amount of money Aria’s car is worth over time is an example of exponential decay. Create a sequence that describes the worth of Aria’s car each year for 10 years. Create a formula for the worth of Aria’s car over time. On a sheet of graph paper, create a graph for the worth of Aria’s car over time. Describe at least one key feature on your graph that signals it depicts exponential decay. Question 6 Explain in your own words how linear functions grow and how exponential functions grow. Explain the statement “No matter what the growth rate of a linear function is, that of an exponential function will always exceed it.” Explain how your linear function from Question 3 and your exponential function from Question 4 demonstrate that this statement is true.

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GPT-4o mini · Answer

### Question 3 **Chosen Amount for Carly's Babysitting:** Let's assume Carly earns **$10** each week for babysitting. **Explanation of Linear Change:** Carly’s savings represent a linear change because she is saving a fixed amount each week. This means that as time progresses (in weeks), her savings increase by a consistent amount ($10) at regular intervals (each week). **Sequence of Savings Over 20 Weeks:** The sequence of her savings over 20 weeks can be described as: - Week 1: $10 - Week 2: $20 - Week 3: $30 - Week 4: $40 - ... - Week 20: $200 This can be expressed as: $$ S(n) = 10n $$ where $ S(n) $ is the savings at week $ n $. **Graphing Carly's Savings:** On graph paper, plot the number of weeks (x-axis) against the savings (y-axis). **Key Feature Indicating Linear Change:** The graph will show a straight line, indicating a constant rate of increase. --- ### Question 4 **Chosen Amount for Nico's Savings Account:** Let’s assume Nico's grandparents put in **$1000** when he was born. **Explanation of Exponential Growth:** The amount in the savings account grows exponentially because the account earns 8% interest annually on the total amount, meaning that the growth each year is a percentage of the current amount, not a fixed amount. Thus, the growth compounds over time. **Sequence of Savings Over 20 Years:** Using compound interest, the amount in the account can be described as follows: - Year 1: $1000 * (1 + 0.08) = $1080 - Year 2: $1000 * (1 + 0.08)² = $1166.40 - Year 3: $1000 * (1 + 0.08)³ = $1259.71 - ... - Year 20: $1000 * (1 + 0.08)²⁰ ≈ $4660.67 This can be expressed as: $$ A(t) = 1000(1.08)^t $$ where $ A(t) $ is the amount in the account after $ t $ years. **Graphing Nico’s Savings:** On graph paper, plot the number of years (x-axis) against the savings (y-axis). **Key Feature Indicating Exponential Growth:** The graph will show a curve that starts off slow but increases rapidly, indicating accelerating growth. --- ### Question 5 **Chosen Amount for Aria's Car:** Let’s assume Aria buys her new car for **$20,000**. **Explanation of Exponential Decay:** The value of Aria’s car depreciates by 15% each year. This means that each year, the car's value is multiplied by a constant factor (0.85), which results in exponential decay since the amount lost each year increases in absolute terms as the car's value decreases. **Sequence of Car Value Over 10 Years:** - Year 1: $20000 * 0.85 = $17,000 - Year 2: $20000 * 0.85² = $14,450 - Year 3: $20000 * 0.85³ = $12,282.50 - ... - Year 10: $20000 * 0.85^{10} ≈ $3,415.24 This can be expressed as: $$ V(t) = 20000(0.85)^t $$ where $ V(t) $ is the value of the car after $ t $ years. **Graphing Aria’s Car Value:** On graph paper, plot the number of years (x-axis) against the car's value (y-axis). **Key Feature Indicating Exponential Decay:** The graph will show a curve that decreases rapidly at first and then level off, indicating decreasing value over time. --- ### Question 6 **Linear vs. Exponential Growth:** - **Linear functions** grow at a constant rate. Regardless of the consistent increment, the growth remains proportional and creates a straight line when graphed. - **Exponential functions**, however, increase at an accelerating rate. The growth increases multiplicatively, meaning larger increments occur as the value grows. **Statement Explanation:** The statement “No matter what the growth rate of a linear function is, that of an exponential function will always exceed it” implies that over time, the effects of compounding in exponential growth will outpace the fixed increments of linear growth. **Demonstration of Statement Using Carly and Nico’s Functions:** - In **Carly’s case**, if she saves $10 weekly, after 20 weeks, she’ll have $200. - In **Nico’s case**, with an initial deposit of $1000 growing at 8%, he’ll have approximately $4660.67 after 20 years. This stark difference between $200 and $4660.67 illustrates how exponential growth can quickly surpass linear growth, even over shorter time frames in the context of different applications. The final amounts clearly demonstrate that Nico’s savings grow faster than Carly’s linear savings trajectory.