Question

directions: fill in the table for the exponential function $y = 2(3)^x - 1$ then select the correct graph and identify the key characteristics.
1.

x	y
-1	-2
0	-1
1	0
2	1

a.
b.
c.
d.
2.
growth / decay
domain:
range:
y-intercept:
asymptote:

directions: write the exponential function equation for the given situation.

1. the mendez family just bought a home for $180,000. if the value of the home increases at a rate of 3% per year, write an exponential function to find the approximate value of the home after 10 years.

$a = 180000$
$r = 3\\%$
$t = 10$
$y = 180,000(1+0.03)^{10}$
bonus: find the value of the home after 10 years:
not solve!

directions: simplify each radical expression.

1. square roots:

a. $sqrt{192}$ =
b. $sqrt{81a^8}$ =
c. $sqrt{72x^9y^{16}}$ =

1. cube roots:

a. $sqrt3{-27} + sqrt{100}$ =
b. $sqrt3{56}$ =
c. $sqrt3{864}$ =

Explanation:

Step1: Verify table values

For $y=2(3)^x -1$:

• $x=-2$: $y=2(3^{-2})-1=2(\frac{1}{9})-1=\frac{2}{9}-1=-\frac{7}{9}\approx-0.78$ (given $-3$ is wrong)
• $x=-1$: $y=2(3^{-1})-1=\frac{2}{3}-1=-\frac{1}{3}\approx-0.33$ (given $-2$ is wrong)
• $x=0$: $y=2(3^0)-1=2(1)-1=1$ (given $-1$ is wrong)
• $x=1$: $y=2(3^1)-1=6-1=5$ (given $0$ is wrong)
• $x=2$: $y=2(3^2)-1=18-1=17$ (given $1$ is wrong)

Step2: Identify graph type

Since base $3>1$, it is exponential growth, so graphs B/D are candidates. Shifted down 1, so asymptote $y=-1$. Correct graph is B.

Step3: Find key characteristics

• Growth/Decay: Growth (base>1)
• Domain: All real numbers ($x\in\mathbb{R}$)
• Range: $y > -1$ (since $2(3)^x>0$, so $2(3)^x-1>-1$)
• y-intercept: $x=0$, $y=1$
• Asymptote: $y=-1$

Step4: Home value calculation

Use $y=a(1+r)^t$, $a=180000$, $r=0.03$, $t=10$:
$y=180000(1+0.03)^{10}=180000(1.03)^{10}$
Calculate $1.03^{10}\approx1.343916$
$y\approx180000\times1.343916=241904.88$

Step5: Simplify square roots

A. $\sqrt{192}=\sqrt{64\times3}=8\sqrt{3}$
B. $\sqrt{81a^8}=9a^4$
C. $\sqrt{72x^9y^{16}}=\sqrt{36x^8y^{16}\times2x}=6x^4y^8\sqrt{2x}$

Step6: Simplify cube roots

A. $\sqrt[3]{-27}+\sqrt{100}=-3+10=7$
B. $\sqrt[3]{56}=\sqrt[3]{8\times7}=2\sqrt[3]{7}$
C. $\sqrt[3]{864}=\sqrt[3]{216\times4}=6\sqrt[3]{4}$

Answer:

1. Corrected table:

x	y
-1	$-\frac{1}{3}$
0	1
1	5
2	17

Correct graph: B
2.

• Growth / Decay: Growth
• Domain: $(-\infty, \infty)$ or all real numbers
• Range: $(-1, \infty)$
• y-intercept: $(0, 1)$
• Asymptote: $y=-1$

1. Exponential function: $y=180000(1+0.03)^{10}$

Bonus: $\$241,904.88$

1. Square roots:

A. $8\sqrt{3}$
B. $9a^4$
C. $6x^4y^8\sqrt{2x}$

1. Cube roots:

A. $7$
B. $2\sqrt[3]{7}$
C. $6\sqrt[3]{4}$