Does a linear function or a nonlinear function represent this situation? Explain.
Understand the Problem
The question asks whether Justin's savings situation can be modeled by a linear function or a nonlinear function, and it requires an explanation based on the information given about his savings pattern.
Answer
The situation can be modeled by a linear function: $$ \text{Money in Account} = 4 + 2 \times \text{Week} $$
Answer for screen readers
Justin's savings situation can be modeled by a linear function, represented by the equation: $$ \text{Money in Account} = 4 + 2 \times \text{Week} $$
Steps to Solve
- Identify Initial Amount and Weekly Savings
Justin starts with $4 and saves an additional $2 each week. This can be represented as:
- Initial amount: $4
- Savings per week: $2
- Establish the Relationship
Each week, the total amount saved increases by $2. This means that for every increase in week (e.g., from week 0 to week 1), the money in the account also increases linearly. The savings can be calculated as: $$ \text{Money in Account} = 4 + 2 \times \text{Week} $$
- Create a Table of Values
From the given information:
- Week 0: $4
- Week 1: $6 (4 + 2)
- Week 2: $8 (4 + 2 \times 2)
- Week 3: $10 (4 + 2 \times 3)
- Week 4: $12 (4 + 2 \times 4)
- Week 5: $14 (4 + 2 \times 5)
This table confirms a consistent increase in savings.
- Determine Function Type
Since the relationship is consistent and can be modeled using the equation derived, Justin's savings situation is represented by a linear function.
Justin's savings situation can be modeled by a linear function, represented by the equation: $$ \text{Money in Account} = 4 + 2 \times \text{Week} $$
More Information
A linear function describes situations where there is a constant rate of change—in this case, Justin saves $2 each week, which results in a straight-line graph when plotted.
Tips
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