Finding the Equation of a Parabolic Arch Bridge

As a data analyst trainee at a manufacturing firm, you are asked to mathematically model a parabolic production cost curve from a visual graph provided by the finance team. To build this model, you need to extract the equation directly from the visual data. Arrange the following steps in the correct order to determine the quadratic function.

A design engineer is modeling the parabolic support of a bridge using a coordinate graph. After identifying the vertex $(h, k)$ and substituting it into the vertex form $f(x) = a(x - h)^2 + k$, what is the next essential step required to determine the value of the leading coefficient $a$?

As a technician at a precision manufacturing firm, you are using the vertex form to model a parabolic lighting reflector based on a 2D graph from the design team. Match each mathematical component to its specific role in creating this model.

An urban planner is modeling the parabolic trajectory of a decorative fountain spray on a coordinate system. To begin deriving the quadratic function for this path from a graph, the planner identifies the coordinates of the fountain's peak and substitutes them into the ______ form of a quadratic function, $f(x) = a(x - h)^2 + k$.

A telecommunications engineer is deriving the mathematical model for a parabolic satellite dish from its coordinate graph. True or False: To fully determine the specific quadratic function $f(x) = a(x - h)^2 + k$, identifying the coordinates of the vertex $(h, k)$ is the only information required from the graph.