1. We are asked to graph the function $f(x) = x^4 - 3x^2 + 2$. 2. The function is a polynomial, so it is continuous and smooth everywhere. 3. To understand the graph, we find critical points by differentiating: $$f'(x) = 4x^3 - 6x$$ 4. Set the derivative equal to zero to find critical points: $$4x^3 - 6x = 0$$ $$2x(2x^2 - 3) = 0$$ 5. Solve for $x$: $$2x = 0 \Rightarrow x = 0$$ $$2x^2 - 3 = 0 \Rightarrow x^2 = \frac{3}{2} \Rightarrow x = \pm \sqrt{\frac{3}{2}}$$ 6. Evaluate $f(x)$ at critical points: $$f(0) = 0^4 - 3\cdot0^2 + 2 = 2$$ $$f\left(\pm \sqrt{\frac{3}{2}}\right) = \left(\sqrt{\frac{3}{2}}\right)^4 - 3\left(\sqrt{\frac{3}{2}}\right)^2 + 2$$ Calculate stepwise: $$\left(\sqrt{\frac{3}{2}}\right)^2 = \frac{3}{2}$$ $$\left(\sqrt{\frac{3}{2}}\right)^4 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ So, $$f\left(\pm \sqrt{\frac{3}{2}}\right) = \frac{9}{4} - 3 \cdot \frac{3}{2} + 2 = \frac{9}{4} - \frac{9}{2} + 2$$ Convert to common denominator 4: $$\frac{9}{4} - \frac{18}{4} + \frac{8}{4} = \frac{9 - 18 + 8}{4} = \frac{-1}{4} = -0.25$$ 7. The function has local minima at $x = \pm \sqrt{\frac{3}{2}}$ with value $-0.25$ and a local maximum at $x=0$ with value $2$. 8. The end behavior as $x \to \pm \infty$ is $f(x) \to +\infty$ because the leading term $x^4$ dominates. 9. The graph crosses the y-axis at $f(0) = 2$. 10. Summary: The graph has a "W" shape with local maxima at $x=0$ and local minima at $x= \pm \sqrt{\frac{3}{2}}$. This analysis helps sketch the graph accurately.