Question

thinking

1. 12 years ago, a house was worth $467000. the value of the house is now $845000. determine the houses appreciation rate assuming that the house increases in value at a constant rate every year. show your steps. t5

1. a. write two other equations that are equivalent to \\(y = 2401^x\\) t2

b. describe any transformations that correspond to the new equations t2

c. state the domain, range and the equation of the horizontal asymptote for each equation. t3

Explanation:

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<reasoning>

Solve for the appreciation rate in Question 10

\[

$$\begin{aligned} V(t) &= P(1 + r)^t \\ 845000 &= 467000(1 + r)^{12} \\ (1 + r)^{12} &= \frac{845000}{467000} \approx 1.80942 \\ 1 + r &= (1.80942)^{\frac{1}{12}} \approx 1.0506 \\ r &\approx 0.0506 \text{ or } 5.06\% \end{aligned}$$

\]

Find equivalent equations for Question 11a

\[

$$\begin{aligned} 2401 &= 7^4 \implies y = (7^4)^x = 7^{4x} \\ 2401 &= 49^2 \implies y = (49^2)^x = 49^{2x} \end{aligned}$$

\]

Describe transformations for Question 11b

\[

$$\begin{aligned} &\text{For } y = 7^{4x}: \text{Horizontal compression by a factor of } \frac{1}{4} \text{ relative to } y = 7^x \\ &\text{For } y = 49^{2x}: \text{Horizontal compression by a factor of } \frac{1}{2} \text{ relative to } y = 49^x \end{aligned}$$

\]

State domain, range, and asymptote for Question 11c

\[

$$\begin{aligned} &\text{Domain: } \{x \in \mathbb{R}\} \\ &\text{Range: } \{y \in \mathbb{R} \mid y > 0\} \\ &\text{Horizontal Asymptote: } y = 0 \end{aligned}$$

\]
</reasoning>

<answer>

Question 10

The house's appreciation rate is approximately \(5.06\%\) per year.

Question 11

a.

Two equivalent equations are:

1. \(y = 7^{4x}\)
2. \(y = 49^{2x}\)

b.

• For \(y = 7^{4x}\), the graph is horizontally compressed by a factor of \(\frac{1}{4}\) compared to the parent function \(y = 7^x\).
• For \(y = 49^{2x}\), the graph is horizontally compressed by a factor of \(\frac{1}{2}\) compared to the parent function \(y = 49^x\).

c.

For all three equivalent equations (\(y = 2401^x\), \(y = 7^{4x}\), and \(y = 49^{2x}\)):

• Domain: \((-\infty, \infty)\)
• Range: \((0, \infty)\)
• Horizontal Asymptote: \(y = 0\)

</answer>

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Answer:

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<reasoning>

Solve for the appreciation rate in Question 10

\[

$$\begin{aligned} V(t) &= P(1 + r)^t \\ 845000 &= 467000(1 + r)^{12} \\ (1 + r)^{12} &= \frac{845000}{467000} \approx 1.80942 \\ 1 + r &= (1.80942)^{\frac{1}{12}} \approx 1.0506 \\ r &\approx 0.0506 \text{ or } 5.06\% \end{aligned}$$

\]

Find equivalent equations for Question 11a

\[

$$\begin{aligned} 2401 &= 7^4 \implies y = (7^4)^x = 7^{4x} \\ 2401 &= 49^2 \implies y = (49^2)^x = 49^{2x} \end{aligned}$$

\]

Describe transformations for Question 11b

\[

$$\begin{aligned} &\text{For } y = 7^{4x}: \text{Horizontal compression by a factor of } \frac{1}{4} \text{ relative to } y = 7^x \\ &\text{For } y = 49^{2x}: \text{Horizontal compression by a factor of } \frac{1}{2} \text{ relative to } y = 49^x \end{aligned}$$

\]

State domain, range, and asymptote for Question 11c

\[

$$\begin{aligned} &\text{Domain: } \{x \in \mathbb{R}\} \\ &\text{Range: } \{y \in \mathbb{R} \mid y > 0\} \\ &\text{Horizontal Asymptote: } y = 0 \end{aligned}$$

\]
</reasoning>

<answer>

Question 10

The house's appreciation rate is approximately \(5.06\%\) per year.

Question 11

a.

Two equivalent equations are:

1. \(y = 7^{4x}\)
2. \(y = 49^{2x}\)

b.

• For \(y = 7^{4x}\), the graph is horizontally compressed by a factor of \(\frac{1}{4}\) compared to the parent function \(y = 7^x\).
• For \(y = 49^{2x}\), the graph is horizontally compressed by a factor of \(\frac{1}{2}\) compared to the parent function \(y = 49^x\).

c.

For all three equivalent equations (\(y = 2401^x\), \(y = 7^{4x}\), and \(y = 49^{2x}\)):

• Domain: \((-\infty, \infty)\)
• Range: \((0, \infty)\)
• Horizontal Asymptote: \(y = 0\)

</answer>

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