Question

the value of some cell phones changes exponentially after initial release. here are graphs showing the depreciation of two phones, 1, 2 and 3 years after they were released. 1. which phone was more expensive to buy when it first came out? which point on the graph tells us this information? 2. which one is falling in value more quickly? how do you know? justify your response with mathematical calculations or reasoning on your whiteboards. 3. generalize and describe how to determine which function will decay more quickly or more slowly when comparing any exponential decay function. be specific in your justification. 4. write an equation to model phone a and phone b.

Explanation:

Step1: Determine the initial price

The initial price of a phone is the value when \(t = 0\). For Phone A, when \(t=0\), \(y = 1000\) (from the point \((0,1000)\)). For Phone B, when \(t = 0\), \(y=840\) (from the point \((0,840)\)).

Step2: Find the decay factor for each phone

The general form of an exponential decay function is \(y=a\cdot b^{t}\), where \(a\) is the initial value and \(b\) is the decay factor.
For Phone A:
We know \(a = 1000\) and when \(t = 1\), \(y=600\). Substitute into \(y=a\cdot b^{t}\):
\(600=1000\cdot b^{1}\), so \(b=\frac{600}{1000}=0.6\)
For Phone B:
We know \(a = 840\) and when \(t = 1\), \(y = 630\). Substitute into \(y=a\cdot b^{t}\):
\(630=840\cdot b^{1}\), so \(b=\frac{630}{840}=0.75\)

Step3: Compare the decay rates

Since \(0.6<0.75\), Phone A has a smaller decay factor. In an exponential decay function \(y=a\cdot b^{t}\) (\(0 < b<1\)), a smaller \(b\) means a faster decay.

Step4: Write the equations

For Phone A: \(y = 1000\cdot(0.6)^{t}\)
For Phone B: \(y=840\cdot(0.75)^{t}\)

Answer:

1. Phone A was more expensive. The point \((0,1000)\) for Phone A and \((0,840)\) for Phone B (the \(y -\) intercepts) tell us this information.
2. Phone A is falling in value more quickly. Using the exponential decay formula \(y=a\cdot b^{t}\), for Phone A \(b = 0.6\) and for Phone B \(b=0.75\). Since \(0.6<0.75\) (and in \(y=a\cdot b^{t}\), \(0 < b<1\) for decay), a smaller \(b\) implies a faster - decaying function.
3. For two exponential decay functions \(y_1=a_1\cdot b_1^{t}\) and \(y_2=a_2\cdot b_2^{t}\) (\(a_1,a_2>0\) and \(0 < b_1,b_2<1\)), compare the decay factors \(b_1\) and \(b_2\). The function with the smaller \(b\) value decays more quickly. For example, if \(b_1 < b_2\), then \(y_1=a_1\cdot b_1^{t}\) decays faster than \(y_2=a_2\cdot b_2^{t}\) because as \(t\) increases, \(b_1^{t}\) gets smaller more rapidly than \(b_2^{t}\).

4.

• Phone A: \(y = 1000\cdot(0.6)^{t}\)
• Phone B: \(y=840\cdot(0.75)^{t}\)