Question

How does adding/subtracting a constant at the end of the function expression affect the function?

Answer 1

Adding or subtracting a constant at the end of a function expression shifts the entire graph vertically. If you add a constant, the graph moves upward; if you subtract a constant, the graph shifts downward.

Step-by-step explanation:

When a constant
`\(c\)` is added to the function
`\(f(x)\)`, it can be expressed as
`\(f(x) + c\).` This addition affects the output of the function for every
`\(x\)` value by adding
`\(c\)` to it. Visually, on a graph, this results in a vertical shift upward by
`\(c\)` units. Mathematically, for any
`\(x\)`,
`\(f(x + c) = f(x) + c\).`

Conversely, subtracting a constant
`\(c\)` from the function
`\(f(x)\)`is represented as
`\(f(x) - c\)`. This subtraction causes the function's output for each
`\(x\)` value to decrease by
`\(c\)`. Graphically, this leads to a vertical shift downward by
`\(c\)` units. Mathematically, for any
`\(x\)`,
`\(f(x - c) = f(x) - c\).`

These transformations are part of the broader field of translation in mathematics. The addition or subtraction of a constant term influences the position of the entire function, but it does not alter its shape or general characteristics.

Understanding these basic transformations is crucial in analyzing and graphing functions, as they provide insights into how changes in the function expression impact its graphical representation.