The equation \(\mathrm{D = 5{,}640(1.9)^t}\) above estimates the global data traffic D, in terabytes, for the year that is t...

1. TRANSLATE the problem information

• Given: \(\mathrm{D = 5,640(1.9)^t}\)
• D = data traffic in terabytes
• t = years after 2010
• Question: What does 5,640 represent?

2. INFER the strategy for interpreting coefficients

• In exponential functions of the form \(\mathrm{f(x) = a(b)^x}\), the coefficient 'a' represents the initial value
• To find what 5,640 represents, evaluate the function when \(\mathrm{t = 0}\)
• Since t represents "years after 2010," \(\mathrm{t = 0}\) corresponds to the year 2010

3. SIMPLIFY by substituting t = 0

• \(\mathrm{D = 5,640(1.9)^0}\)
• Since any number to the 0 power equals 1: \(\mathrm{(1.9)^0 = 1}\)
• \(\mathrm{D = 5,640(1) = 5,640}\)

4. INFER the meaning

• When \(\mathrm{t = 0}\) (year 2010), \(\mathrm{D = 5,640}\) terabytes
• Therefore, 5,640 represents the estimated data traffic in 2010

Answer: D. The estimated data traffic, in terabytes, in 2010

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to evaluate the function at \(\mathrm{t = 0}\) to understand what the coefficient represents. Instead, they might think 5,640 represents the amount of increase each year or confuse it with the growth rate. This conceptual confusion about how exponential functions work may lead them to select Choice A (yearly increase) or Choice B (percent increase).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "t years after 2010" means and don't realize that \(\mathrm{t = 0}\) gives them the 2010 value. They might think 5,640 represents the entire expression for any year, leading them to select Choice C (general traffic formula).

The Bottom Line:

Understanding exponential functions requires recognizing that the coefficient represents the initial value when the exponent equals zero. Students who don't make this connection will struggle to interpret what each part of the equation represents.