1. **Stating the problem:** We want to understand how changing the coefficients $a$, $b$, and $c$ in the quadratic equation $y = ax^2 + bx + c$ affects the discriminant $D = b^2 - 4ac$. 2. **Formula and explanation:** The discriminant $D = b^2 - 4ac$ determines the nature of the roots of the quadratic equation. - If $D > 0$, there are two distinct real roots. - If $D = 0$, there is exactly one real root (a repeated root). - If $D < 0$, there are no real roots (complex roots). 3. **Given values:** $a = 1$, $b = 4$, $c = 1$, so $$D = 4^2 - 4 \times 1 \times 1 = 16 - 4 = 12$$ 4. **Effect of changing $a$:** - Since $D = b^2 - 4ac$, increasing $a$ increases the term $4ac$, which subtracts more from $b^2$, thus decreasing $D$. - Decreasing $a$ decreases $4ac$, increasing $D$. 5. **Effect of changing $b$:** - $D$ depends on $b^2$, so increasing the absolute value of $b$ increases $b^2$, making $D$ larger. - Decreasing $|b|$ makes $b^2$ smaller, decreasing $D$. 6. **Effect of changing $c$:** - Increasing $c$ increases $4ac$, decreasing $D$. - Decreasing $c$ decreases $4ac$, increasing $D$. 7. **Summary:** - To make $D$ larger, increase $|b|$ or decrease $a$ or $c$. - To make $D$ smaller, decrease $|b|$ or increase $a$ or $c$. This means you can change any of $a$, $b$, or $c$ to affect $D$, but $b$ affects $D$ quadratically, while $a$ and $c$ affect it linearly. **Final answer:** Changing $b$ (especially its magnitude) has the strongest effect on increasing $D$, while increasing $a$ or $c$ decreases $D$.