Question

analyzing rates of change

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$$\begin{array}{|c|c|} \\hline \\text{years} & \\text{value (\\$)} \\\\ \\hline 1 & 5,250 \\\\ \\hline 2 & 5,512.50 \\\\ \\hline 3 & 5,788.13 \\\\ \\hline 4 & 6,077.54 \\\\ \\hline 5 & 6,381.42 \\\\ \\hline 6 & 6,700.49 \\\\ \\hline \\end{array}$$

the data in the table represents the value of a savings account at the end of each year for 6 years. the relationship between the increasing years and the increasing value of the account is exponential.

there is dropdown rate of change in an exponential relationship.

after each year, the value of the account is dropdown times as large as the previous year.

Explanation:

Analyze the rate of change in an exponential relationship

Using the Average Rate of Change knowledge point
An exponential relationship does not grow by a constant amount each year; instead, it grows by a constant percentage or factor. Therefore, the absolute rate of change is not constant, but rather changing (specifically, increasing for growth).
Thus, there is a variable (or a changing) rate of change in an exponential relationship.

Calculate the growth factor between consecutive years

Using the Common Ratio knowledge point

$$ \text{Ratio} = \frac{\text{Value in Year 2}}{\text{Value in Year 1}} = \frac{5512.50}{5250} = 1.05 $$

$$ \text{Ratio} = \frac{\text{Value in Year 3}}{\text{Value in Year 2}} = \frac{5788.13}{5512.50} \approx 1.05 $$

The constant multiplier (growth factor) is \(1.05\).

Answer:

The data in the table represents the value of a savings account at the end of each year for 6 years. The relationship between the increasing years and the increasing value of the account is exponential.

There is <blank>a variable</blank> rate of change in an exponential relationship.

After each year, the value of the account is <blank>1.05</blank> times as large as the previous year.