A function is given algebraically by the formula...

A function is given algebraically by the formula $f(x)=(x-4)^{2}+3 .$ Complete these other ways to represent $f:$
a. Verbal: "Subtract 4, then ____ and ____.
b. Numerical:
$$\begin{array}{|c|c|}
\hline x & f(x) \\
\hline 0 & 19 \\
2 & \\
4 & \\
6 & \\
\hline
\end{array}$$

Key Concepts

Order of Operations

Order of operations is the set of rules that determines the sequence in which operations are performed in an expression. In the context of functions, it ensures that when applying multiple operations—such as subtraction, exponentiation, and addition—the correct output is obtained, maintaining the integrity of the function's structure.

Function Evaluation

Function evaluation is the process of computing the output of a function for a given input by substituting the input into the function’s formula and following the prescribed operations. This key concept underpins numerical representations of functions and is crucial for analyzing and graphing functions.

Function Representation

Function representation is the practice of expressing the relationship between variables in different forms such as algebraic, verbal, numerical, and graphical. This concept is important because it enables an understanding of the behavior of functions from various perspectives, making it easier to analyze and communicate mathematical ideas.

Transformation of Functions

Transformation of functions involves modifying a basic function to produce a new function by applying shifts, stretches, reflections, or compressions. Recognizing these transformations, such as horizontal and vertical shifts, is essential for understanding how changes to the algebraic expression affect the shape and position of the function's graph.

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