Exponential Graph 4B89A7

1. **State the problem:** We need to analyze and graph the exponential function $$f(x) = -6\left(\frac{4}{3}\right)^x$$ and answer questions about its behavior. 2. **Recall the general form and properties:** An exponential function has the form $$f(x) = a b^x$$ where $a$ is a constant and $b > 0$ is the base. - If $a < 0$ and $b > 1$, the function is decreasing. - The domain of any exponential function is all real numbers: $$(-\infty, \infty)$$. - The range depends on $a$; since $a$ is negative, the function values will be negative. - The horizontal asymptote is $y=0$. 3. **Determine if the function is increasing or decreasing:** Since $a = -6 < 0$ and $b = \frac{4}{3} > 1$, the function is **decreasing**. 4. **Find the domain and range:** - Domain: $$(-\infty, \infty)$$ because exponential functions are defined for all real $x$. - Range: Since $a$ is negative, the function outputs negative values only, so the range is $$(-\infty, 0)$$. 5. **Find the y-intercept:** The y-intercept occurs at $x=0$: $$f(0) = -6\left(\frac{4}{3}\right)^0 = -6 \times 1 = -6$$ 6. **Find the horizontal asymptote:** As $x \to \infty$, $$\left(\frac{4}{3}\right)^x \to \infty$$, but multiplied by $-6$ it goes to $-\infty$. As $x \to -\infty$, $$\left(\frac{4}{3}\right)^x \to 0^+$$, so $$f(x) \to -6 \times 0 = 0$$ from below. Thus, the horizontal asymptote is $$y = 0$$. 7. **Summary:** - The function is decreasing. - Domain: $$(-\infty, \infty)$$. - Range: $$(-\infty, 0)$$. - Y-intercept: $-6$. - Horizontal asymptote: $y=0$. 8. **Graph shape:** The graph is an exponential decay curve below the x-axis, approaching $y=0$ from below on the left and decreasing without bound on the right.