Question

1. $y=\frac{1}{2}\cdot2^{x}-4$

growth / decay
domain: ________
range: ________
y-intercept: ________
asymptote: ________

Explanation:

Step1: Identify growth/decay

The function is $y=\frac{1}{2}\cdot2^x - 4$. The base of the exponential term is $2>1$, so it represents exponential growth.

Step2: Find the domain

Exponential functions accept all real numbers as input, so there are no restrictions on $x$.
Domain: All real numbers, or $(-\infty, \infty)$

Step3: Calculate the range

The term $2^x$ has a range of $(0, \infty)$. Multiply by $\frac{1}{2}$: $\frac{1}{2}\cdot2^x \in (0, \infty)$. Subtract 4: $\frac{1}{2}\cdot2^x - 4 \in (-4, \infty)$

Step4: Find y-intercept

Set $x=0$ and solve for $y$:
$y=\frac{1}{2}\cdot2^0 - 4 = \frac{1}{2}\cdot1 - 4 = \frac{1}{2} - 4 = -\frac{7}{2}$

Step5: Identify asymptote

For $y = a\cdot b^x + k$, the horizontal asymptote is $y=k$. Here $k=-4$, so the asymptote is $y=-4$.

Answer:

Growth / Decay: Growth
Domain: $(-\infty, \infty)$
Range: $(-4, \infty)$
y-intercept: $\boldsymbol{-\frac{7}{2}}$ (or $-3.5$)
Asymptote: $\boldsymbol{y=-4}$