1. **State the problem:** We are given the function $$y = -\frac{1}{128} \cdot 4^{3(x+2)} + 30$$ and want to understand its behavior and graph. 2. **Recall the general form:** The function is of the form $$y = a \cdot c^{x} + k$$ where: - $a$ is the vertical stretch/compression and reflection factor, - $c$ is the base of the exponential, - $k$ is the vertical shift. 3. **Identify parameters:** Here, - $a = -\frac{1}{128}$ (negative, so reflection over x-axis and very small magnitude), - $c = 4^{3} = 64$ because $4^{3(x+2)} = (4^3)^{x+2} = 64^{x+2}$, - $k = 30$ (vertical shift upwards by 30). 4. **Rewrite the function:** $$y = -\frac{1}{128} \cdot 64^{x+2} + 30$$ 5. **Analyze behavior:** - As $x \to \infty$, $64^{x+2} \to \infty$, so $y \to -\infty$ because of the negative coefficient. - As $x \to -\infty$, $64^{x+2} \to 0$, so $y \to 30$ from below. 6. **Find y-intercept:** Set $x=0$: $$y = -\frac{1}{128} \cdot 64^{2} + 30 = -\frac{1}{128} \cdot 4096 + 30 = -32 + 30 = -2$$ 7. **Horizontal asymptote:** $y = 30$ **Final answer:** The function is an exponential decay reflected over the x-axis, shifted up by 30, with horizontal asymptote $y=30$ and y-intercept at $(0,-2)$.