Question 125·Medium·Nonlinear Functions
A collector bought an antique vase on January 1. The function , where , gives the predicted price, in dollars, of the vase years after the purchase.
What is the best interpretation of the statement " is approximately equal to " in this context?
For function-interpretation questions, always start by restating the definition: what does the input variable represent (time, number of items, etc.) and what does the function output represent (price, distance, etc.)? Then plug in the given input (here, ), translate that input into words, and describe the output as the corresponding quantity at that time or amount. Finally, eliminate answer choices that talk about changes, comparisons, or totals (like "increase", "percent greater", or "times as much") unless the expression is explicitly a difference or ratio, such as or . This keeps you focused on what the function value itself means.
Hints
Identify input and output
Carefully reread the sentence that defines . What does the input represent in this situation? What does the output represent?
Interpret the specific input value
If is the number of years after the vase was bought, what moment in time does describe in words?
Focus on what is, not how it changes
Does tell you how much the price changed since the previous year, or does it tell you the actual price at a specific time? Use this to eliminate choices that talk about increases or comparisons instead of the price itself.
Desmos Guide
Enter the function into Desmos
Type p(t) = 120*(1.07)^t into Desmos so the graph or expression for the predicted price is defined.
Evaluate the function at 5 years
In Desmos, either type p(5) or 120*(1.07)^5 and read off the output value. Then think: this output is the predicted price, and the input 5 represents 5 years after the vase was purchased, so interpret that pair in words.
Step-by-step Explanation
Understand what the function represents
The problem says "gives the predicted price, in dollars, of the vase years after the purchase."
So:
- The input is time in years after January 1 when the vase was bought.
- The output is the predicted price of the vase in dollars at that time.
Interpret in words
Now focus on .
Since is the number of years after the purchase date, means 5 years after the vase was purchased.
Therefore, represents the vase’s predicted price 5 years after the purchase (in dollars).
Connect to the context and match the choice
The statement " is approximately equal to 168" means that, at years after purchase, the model predicts a price of about 168 dollars.
In words, this is:
5 years after the vase was purchased, its predicted price is approximately 168 dollars.
That matches answer choice D. The other options add extra ideas (percent increase, being a multiple of a previous price, or total increase) that are not what itself represents.