Graph stuff - 2025-03-26

Similarities:

• All are quadratic expressions in the form $a{x}^2+bx+c$.
• Degree 2, with variable $x$.
• Real Coefficients: All coefficients ($a$, $b$, and $c$) are real numbers.

Differences:

1. Coefficients:

   • $4x^2-12x-40$: $a=4$, $b=-12$, $c=-40$
   • $-2x^2+6x+20$: $a=-2$, $b=6$, $c=20$
   • $x^2-3x-10$: $a=1$, $b=-3$, $c=-10$

2. Direction of Parabola:

   • Opens up: $4x^2-12x-40$, $x^2-3x-10$
   • Opens down: $-2x^2+6x+20$

3. Roots:
   Roots will vary based on the discriminant $\Delta =b^2-4ac$.

$$\begin{array}{rl}& \text{Summary of Roots:}\\ & \\ & \text{For }4x^2-12x-40:x=5,-2\\ & \text{For }-2x^2+6x+20:x=-2,5\\ & \text{For }{x}^2-3x-10:x=5,-2\end{array}$$