which equation could generate the curve in the graph below?\n$y = 9x^{2}+6x + 4$\n$y = 6x^{2}-12x - 6$\n$y =…

Explanation:

Step1: Identify vertex property

The parabola has its vertex on the x-axis, so the quadratic has a repeated root (discriminant = 0). The discriminant of $y=ax^2+bx+c$ is $\Delta=b^2-4ac$.

Step2: Calculate discriminant for each option

1. For $y=9x^2+6x+4$: $\Delta=6^2-4(9)(4)=36-144=-108\neq0$
2. For $y=6x^2-12x-6$: $\Delta=(-12)^2-4(6)(-6)=144+144=288\neq0$
3. For $y=3x^2+7x+5$: $\Delta=7^2-4(3)(5)=49-60=-11\neq0$
4. For $y=2x^2+8x+8$: $\Delta=8^2-4(2)(8)=64-64=0$

Step3: Verify vertex position

Find the x-coordinate of the vertex of $y=2x^2+8x+8$ using $x=-\frac{b}{2a}$: $x=-\frac{8}{2(2)}=-2$ Substitute $x=-2$ into the equation: $y=2(-2)^2+8(-2)+8=8-16+8=0$ The vertex is $(-2,0)$, which lies on the x-axis, matching the graph.

Answer:

$y = 2x^2 + 8x + 8$