1. **State the problem:** We are given the exponential function $$y = \frac{1}{2} \cdot 4^x$$ and need to analyze its growth/decay, domain, range, and y-intercept. 2. **Formula and rules:** The general form of an exponential function is $$y = a \cdot b^x$$ where: - $a$ is the initial value (y-intercept). - $b$ is the base; if $b > 1$, the function shows exponential growth; if $0 < b < 1$, it shows exponential decay. - The domain of any exponential function is all real numbers $\mathbb{R}$. - The range depends on $a$ and $b$ but for $a > 0$, the range is $(0, \infty)$. 3. **Analyze the given function:** - Here, $a = \frac{1}{2}$ and $b = 4$. - Since $b = 4 > 1$, the function exhibits exponential growth. - The domain is all real numbers: $$\text{Domain} = \mathbb{R}$$ - The y-intercept occurs at $x=0$: $$y = \frac{1}{2} \cdot 4^0 = \frac{1}{2} \cdot 1 = \frac{1}{2} = 0.5$$ 4. **Find the range:** - Since $a > 0$ and $b > 1$, the function is always positive and increases without bound as $x$ increases. - As $x \to -\infty$, $4^x \to 0$, so $y \to 0$ from above. - As $x \to \infty$, $y \to \infty$. - Therefore, the range is: $$\text{Range} = (0, \infty)$$ 5. **Graph shape description:** - The graph starts near zero for large negative $x$ values. - It passes through the y-intercept at $(0, 0.5)$. - It increases steeply as $x$ becomes positive, showing exponential growth. **Final summary:** - $$y = \frac{1}{2} \cdot 4^x$$ is an exponential growth function. - Domain: $$\mathbb{R}$$ - Range: $$(0, \infty)$$ - Y-intercept: $0.5$