You are a financial analyst reviewing a parabolic profit curve for a new product line. To forecast future earnings, you need to model this curve using the vertex form $f(x) = a(x-h)^2 + k$. According to the process for determining a quadratic function from its graph, which specific pieces of information must you identify first to use this formula?

A facility manager is using a parabolic model to predict the path of a water sprinkler to ensure optimal lawn coverage. To determine the exact quadratic function $f(x) = a(x-h)^2 + k$ from a plotted spray graph, the manager must follow a standard procedure. Arrange the following steps in the correct order to find the quadratic function.

An engineering firm is modeling the parabolic support cable for a bridge using the vertex form of a quadratic function: $f(x) = a(x-h)^2 + k$. Match each part of the formula or modeling process with its correct description.

Cooling Tower Efficiency Analysis

When determining a quadratic function from its graph using the vertex form $f(x) = a(x-h)^2 + k$, the coordinates of the $y$-intercept can be used as the secondary point to solve for the leading coefficient $a$.

Modeling Daily Retail Foot-Traffic

An inventory manager is analyzing a parabolic graph that represents warehouse stock levels over a specific period. To determine the exact quadratic function using the vertex form $f(x) = a(x-h)^2 + k$, the manager first identifies the vertex. To solve for the leading coefficient $a$, the manager must then identify a ____ point on the graph, such as the $y$-intercept, and substitute its coordinates into the equation.