1. **State the problem:** Write a quadratic function in factored form with x-intercepts $\left(-\frac{1}{4}, 0\right)$ and $(8, 0)$ that passes through the point $(0, 2)$. 2. **Recall the factored form of a quadratic:** If the roots are $r_1$ and $r_2$, the quadratic can be written as $$y = a(x - r_1)(x - r_2)$$ where $a$ is a constant. 3. **Substitute the roots:** Here, $r_1 = -\frac{1}{4}$ and $r_2 = 8$, so $$y = a\left(x - \left(-\frac{1}{4}\right)\right)(x - 8) = a\left(x + \frac{1}{4}\right)(x - 8)$$ 4. **Use the point $(0, 2)$ to find $a$:** Substitute $x=0$ and $y=2$: $$2 = a\left(0 + \frac{1}{4}\right)(0 - 8) = a \cdot \frac{1}{4} \cdot (-8) = a \cdot \left(-2\right)$$ 5. **Solve for $a$:** $$2 = -2a \implies a = \frac{2}{-2} = -1$$ 6. **Write the final equation:** $$y = -1 \left(x + \frac{1}{4}\right)(x - 8) = -\left(x + \frac{1}{4}\right)(x - 8)$$ This is the quadratic function in factored form that meets the conditions.