1. **State the problem:** We are given four variables $N_1$, $N_2$, $N_3$, and $V_4$ defined by algebraic expressions involving $f$, $g$, and $h$, which themselves depend on $\phi$ and other parameters. The task is to plot $V$ versus $\phi$, where $V$ corresponds to $V_4$. 2. **Given expressions:** $$N_1 = \sum f^{ch}_3$$ $$N_2 = 1 - 8 m_0 f^{3/2}$$ $$N_3 = \frac{1}{2} f^2$$ $$V_4 = 1 - 12 C g^2 f^{ch 3/2}$$ where $$f = (\sqrt{4 \tau + 4 W \phi - c''})^2 = 4 \tau + 4 W \phi - c''$$ $$g = \left(\frac{1}{2} - B \phi\right)^2$$ $$h = \left(\tanh\left(\frac{\sqrt{f}}{4 c}\right)\right)^2$$ 3. **Focus on $V_4$ as $V$ to plot:** We rewrite $V_4$ as $$V_4 = 1 - 12 C g^2 f^{3/2}$$ 4. **Explain the components:** - $f$ is a quadratic function in $\phi$ (linear inside the square root squared simplifies to linear). - $g$ is a quadratic function in $\phi$. - $f^{3/2} = f^{1.5}$ means we take $f$ to the power $1.5$. 5. **Summary formula for plotting:** $$V(\phi) = 1 - 12 C \left(\frac{1}{2} - B \phi\right)^4 \left(4 \tau + 4 W \phi - c''\right)^{1.5}$$ 6. **Interpretation:** - $V$ depends on $\phi$ through polynomial and power functions. - Parameters $B$, $C$, $\tau$, $W$, and $c''$ are constants that affect the shape. 7. **Plotting note:** - To plot $V$ vs $\phi$, choose a range for $\phi$ where the expression inside the power is non-negative (since $f$ must be $\geq 0$ for real powers). 8. **Desmos-ready function:** $$y = 1 - 12 C \left(\frac{1}{2} - B x\right)^4 \left(4 \tau + 4 W x - c''\right)^{1.5}$$ where $x$ represents $\phi$. This completes the analysis and formula for plotting $V$ versus $\phi$.