1. **Problem Statement:** Analyze the polynomial function $$f(x) = 6x^4 - 23x^3 + 7x^2 + 27x - 9$$ including its zeros, intercepts, extrema, intervals of increase/decrease, and end behavior. 2. **Formula and Rules:** For polynomial functions, zeros are values of $x$ where $f(x) = 0$. The multiplicity of a zero affects the graph's behavior at that root. Relative maxima and minima occur where the first derivative $f'(x)$ changes sign. Intervals of increase/decrease are determined by the sign of $f'(x)$. End behavior depends on the leading term's degree and coefficient. 3. **Zeros and Multiplicities:** Given zeros are $x = -0.848, 0.406, 1.000, 3.441$ each with multiplicity 1, meaning the graph crosses the x-axis at these points. 4. **Intercepts:** The y-intercept is at $(0, -9)$, found by evaluating $f(0) = -9$. 5. **Relative Extrema:** Relative maxima at $(2.798, 25.152)$ and relative minima at $(-0.590, -19.153)$ and $(0.667, -2.271)$ are points where the slope changes from positive to negative or vice versa. 6. **Absolute Extrema:** The absolute minimum is at $(-0.590, -19.153)$; no absolute maximum exists as the function tends to infinity. 7. **Intervals of Increase and Decrease:** The function increases on $(-0.590, 0.667)$ and $(2.798, \, \infty)$ and decreases on $(-\infty, -0.590)$ and $(0.667, 2.798)$. 8. **End Behavior:** As $x \to -\infty$, $f(x) \to \infty$ and as $x \to \infty$, $f(x) \to \infty$ due to the positive leading coefficient and even degree. 9. **Summary Sentence:** The graph of $f(x) = 6x^4 - 23x^3 + 7x^2 + 27x - 9$ has four distinct zeros, relative extrema, and intervals of increase and decrease as described, with end behavior tending to positive infinity on both ends. **Final answer:** The function's key features are zeros at $-0.848, 0.406, 1.000, 3.441$, y-intercept at $(0, -9)$, relative maxima and minima as given, and increasing/decreasing intervals as stated.