1. **State the problem:** We are given the function $f(x) = 4x^4 - 8$ and want to understand its properties. 2. **Formula and rules:** This is a quartic polynomial function of the form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$. Here, $a=4$, $b=0$, $c=0$, $d=0$, and $e=-8$. 3. **Find intercepts:** - **y-intercept:** Set $x=0$, then $f(0) = 4(0)^4 - 8 = -8$. - **x-intercepts:** Set $f(x)=0$: $$4x^4 - 8 = 0$$ $$4x^4 = 8$$ $$\cancel{4}x^4 = \cancel{4}2$$ $$x^4 = 2$$ $$x = \pm \sqrt[4]{2}$$ 4. **Find extrema:** - Take the derivative: $$f'(x) = 16x^3$$ - Set $f'(x) = 0$: $$16x^3 = 0$$ $$x = 0$$ - Second derivative test: $$f''(x) = 48x^2$$ $$f''(0) = 0$$ (inconclusive, but since $f(x)$ is a quartic with positive leading coefficient, $x=0$ is a minimum). 5. **Summary:** - The graph has a minimum at $(0, -8)$. - It crosses the y-axis at $(0, -8)$. - It crosses the x-axis at $x = \pm \sqrt[4]{2}$. - The function grows large positively as $x \to \pm \infty$. **Final answer:** The function $f(x) = 4x^4 - 8$ has x-intercepts at $x = \pm \sqrt[4]{2}$ and a minimum point at $(0, -8)$.