1. **Stating the problem:** We are given the function $$y = c_1 e^{-2x} + c_2 e^{3x}$$ and want to understand its behavior and graph. 2. **Formula and explanation:** This function is a linear combination of two exponential functions: $$e^{-2x}$$ which decreases as $x$ increases, and $$e^{3x}$$ which increases as $x$ increases. 3. **Important rules:** - The term $$e^{-2x}$$ represents exponential decay because the exponent is negative. - The term $$e^{3x}$$ represents exponential growth because the exponent is positive. - Constants $$c_1$$ and $$c_2$$ scale these terms and affect the overall shape. 4. **Intermediate work:** - For large negative $x$, $$e^{-2x}$$ grows large (since $$-2x$$ becomes positive), and $$e^{3x}$$ approaches zero. - For large positive $x$, $$e^{-2x}$$ approaches zero, and $$e^{3x}$$ grows large. 5. **Interpretation:** The graph shows two curves: - The first curve $$c_1 e^{-2x}$$ decreases from the top-left to the right. - The second curve $$c_2 e^{3x}$$ increases from the bottom-left to the right. The overall function $$y$$ is their sum, combining decay and growth behaviors. **Final answer:** The function $$y = c_1 e^{-2x} + c_2 e^{3x}$$ is a sum of exponential decay and growth terms, with constants $$c_1$$ and $$c_2$$ controlling their influence.