Exponential Functions from Situations Quick Check

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Question
Callie entered an art contest in second grade and won a $1,000 scholarship. The money was invested in an account paying a 9% interest rate compounded annually. The situation can be modeled by the equation a(t)=1,000(1.09)t, where a is the amount in the account after t years. If Callie uses the scholarship 10 years later, determine which graph accurately displays the situation.(1 point)
Responses

An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 14 in increments of 2 and the y-axis ranges from 0 to 3,500 in increments of 500.
Image with alt text: An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 14 in increments of 2 and the y-axis ranges from 0 to 3,500 in increments of 500.

An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 5 in increments of 1 and the y-axis ranges from 0 to 2000 in increments of 500.
Image with alt text: An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 5 in increments of 1 and the y-axis ranges from 0 to 2000 in increments of 500.

An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 10 in increments of 1 and the y-axis ranges from 0 to 2,000 in increments of 500.
Image with alt text: An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 10 in increments of 1 and the y-axis ranges from 0 to 2,000 in increments of 500.

A decreasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 12 in increments of 2 and the y-axis ranges from 0 to 1,200 in increments of 200.
Image with alt text: A decreasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 12 in increments of 2 and the y-axis ranges from 0 to 1,200 in increments of 200.
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Answer 1

To determine the correct graph that accurately displays Callie's investment situation, we need to analyze the function provided:

The function modeling her account balance is given as: \[ a(t) = 1000(1.09)^t \]

Here, \( a(t) \) represents the amount in the account after \( t \) years, with the following conditions:

• The account starts with $1,000.
• It is compounded annually at a rate of 9%.

To find the value in the account after 10 years, we can calculate:

\[ a(10) = 1000(1.09)^{10} \]

Now, let's compute this value:

1. Calculate \( (1.09)^{10} \): \[ (1.09)^{10} \approx 2.36736 \]

2. Multiply by $1,000: \[ a(10) \approx 1000 \times 2.36736 \approx 2367.36 \]

So after 10 years, the amount in the account is approximately $2,367.36.

Now let’s compare this with the provided graph options:

1. The first option indicates ranges up to $3,500.
2. The second option only goes up to $2,000, which is not sufficient.
3. The third option goes up to $2,000 but provides an increment of 1 on the x-axis from 0 to 10.
4. The last option depicts a decreasing curve, which is incorrect since the function is increasing due to the compound interest.

Given the calculations and comparison of the values, the third option, which shows an increasing curve plotted from 0 to 10 years on the x-axis and an appropriate scale on the y-axis (up to $2,000) to likely include the value around $2,367.36, is the best choice.

Therefore, the correct response is: An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 10 in increments of 1 and the y-axis ranges from 0 to 2,000 in increments of 500.