Final answer:
The quadratic equation with a vertex at (0, -7), where the vertex is a minimum and following the multiplier pattern (-2, -8, -18), is y = -2x^2 - 7. This is derived using the vertex form of a quadratic equation and analyzing the multiplier pattern to determine the 'a' value.
Step-by-step explanation:
To write a quadratic equation with a vertex at (0,-7), knowing that the vertex is a minimum, and following the multiplier pattern (-2,-8, -18), we use the vertex form of a quadratic equation, which is y = a(x - h)^2 + k, where (h,k) is the vertex of the parabola. Since the vertex is at the origin (0,0) in this case and the y-coordinate of the vertex is -7 (k = -7), we can partially fill in this equation with y = a(x - 0)^2 - 7, which simplifies to y = ax^2 - 7.
To determine the value of a, we can look at the multiplier pattern provided. The pattern (-2,-8, -18) can be seen as the differences between consecutive y-values for consecutive integer x-values. Looking at these differences: from -2 to -8, we have a difference of -6, and from -8 to -18, the difference is -10, which suggests that the quadratic is opening downwards (since the differences are negative and decreasing) and that we're incrementing by 4 each time (as -6 + (-4) equals -10, which can be the next difference if we continue the pattern). This tells us that the coefficient of the x^2 term in the original equation with factor 1 would be reduced by a factor of 2 every time. Hence, based on the multiplier pattern, we see that a should be -2. The completed quadratic equation is y = -2x^2 - 7.