Question

name:
date:
block:
topic 1: graphing exponential & logarithmic functions
describe as an exponential growth or decay.

1. $f(x)=5(\frac{2}{3})^{x}$
2. $f(x)=\frac{1}{3}(\frac{6}{5})^{x}$

graph each function and identify its key characteristics.

1. $f(x)=3^{x + 1}-6$

domain:
range:
end behavior:
as $x\to\infty$, $f(x)\to$
as $x\to-\infty$, $f(x)\to$
y - intercept:
asymptote:

1. $f(x)=(\frac{1}{2})^{x - 5}+2$

domain:
range:
end behavior:
as $x\to\infty$, $f(x)\to$
as $x\to-\infty$, $f(x)\to$
y - intercept:
asymptote:

1. $f(x)=\log_{2}x-3$

domain:
range:
end behavior:
as $x\to$ , $f(x)\to\infty$
as $x\to$ , $f(x)\to-\infty$
x - intercept:
asymptote:

Explanation:

Step1: Determine exponential growth/decay

For an exponential function \(y = a(b)^x\), if \(0 < b<1\), it is decay; if \(b > 1\), it is growth.

• For \(f(x)=5(\frac{2}{3})^x\), since \(0<\frac{2}{3}<1\), it is exponential decay.
• For \(f(x)=\frac{1}{3}(\frac{6}{5})^x\), since \(\frac{6}{5}>1\), it is exponential growth.

Step2: Analyze \(f(x)=3^{x + 1}-6\)

• Domain: The domain of an exponential function \(y = a(b)^{x + h}+k\) is all real numbers, so \(x\in(-\infty,\infty)\).
• Range: The range of \(y = 3^{x+1}\) is \((0,\infty)\). After subtracting 6, the range is \(y\in(-6,\infty)\).
• End - behavior: As \(x\to\infty\), \(3^{x + 1}\to\infty\), so \(f(x)\to\infty\). As \(x\to-\infty\), \(3^{x + 1}\to0\), so \(f(x)\to - 6\).
• y - intercept: Set \(x = 0\), \(f(0)=3^{0 + 1}-6=3 - 6=-3\).
• Asymptote: The horizontal asymptote is \(y=-6\).

Step3: Analyze \(f(x)=(\frac{1}{2})^{x - 5}+2\)

• Domain: All real numbers, \(x\in(-\infty,\infty)\).
• Range: The range of \(y = (\frac{1}{2})^{x-5}\) is \((0,\infty)\). After adding 2, the range is \(y\in(2,\infty)\).
• End - behavior: As \(x\to\infty\), \((\frac{1}{2})^{x - 5}\to0\), so \(f(x)\to2\). As \(x\to-\infty\), \((\frac{1}{2})^{x - 5}\to\infty\), so \(f(x)\to\infty\).
• y - intercept: Set \(x = 0\), \(f(0)=(\frac{1}{2})^{-5}+2=32 + 2=34\).
• Asymptote: The horizontal asymptote is \(y = 2\).

Step4: Analyze \(f(x)=\log_2x-3\)

• Domain: For \(y=\log_bx\), \(x>0\), so \(x\in(0,\infty)\).
• Range: All real numbers, \(y\in(-\infty,\infty)\).
• End - behavior: As \(x\to\infty\), \(\log_2x\to\infty\), so \(f(x)\to\infty\). As \(x\to0^{+}\), \(\log_2x\to-\infty\), so \(f(x)\to-\infty\).
• x - intercept: Set \(y = 0\), \(\log_2x-3=0\Rightarrow\log_2x=3\Rightarrow x = 8\).
• Asymptote: The vertical asymptote is \(x = 0\).

Answer:

1. Exponential decay
2. Exponential growth

3.

• Domain: \((-\infty,\infty)\)
• Range: \((-6,\infty)\)
• End Behavior: As \(x\to\infty\), \(f(x)\to\infty\); As \(x\to-\infty\), \(f(x)\to - 6\)
• y - intercept: \(-3\)
• Asymptote: \(y=-6\)

4.

• Domain: \((-\infty,\infty)\)
• Range: \((2,\infty)\)
• End Behavior: As \(x\to\infty\), \(f(x)\to2\); As \(x\to-\infty\), \(f(x)\to\infty\)
• y - intercept: \(34\)
• Asymptote: \(y = 2\)

5.

• Domain: \((0,\infty)\)
• Range: \((-\infty,\infty)\)
• End Behavior: As \(x\to\infty\), \(f(x)\to\infty\); As \(x\to0^{+}\), \(f(x)\to-\infty\)
• x - intercept: \(8\)
• Asymptote: \(x = 0\)