Transformations and Relationships of Functions

How are the functions y = x and y = x - 3 related? How are their graphs related?

If the plane leaves two hours late, what function represents this transformation of the projected altitude f(x) (in thousands of feet)?

Let g(x) be the reflection of f(x) = 3x + 2 in the x-axis. What is a function rule for g(x)?

The function f(x) is represented by the table below. What does it represent?

What transformations change the graph of f(x) to the graph of g(x)?

Relationships Between Functions

The functions y = x and y = x - 3 are linear and share the same slope, indicating parallel lines.
The graph of y = x - 3 is a vertical shift downwards by 3 units from the graph of y = x.

Function Transformation

For an airplane's altitude graph f(x), a departure time shift by two hours later corresponds to the function being altered to reflect this time change.
The new function will adjust to account for delayed altitude projections, typically represented as f(x - 2).

Reflected Functions

The function g(x) is the reflection of f(x) = 3x + 2 across the x-axis.
The reflection can be expressed mathematically as g(x) = -f(x) = -3x - 2.

Function Representation

f(x) can be represented in various forms, such as a table, which provides numerical values corresponding to specific inputs.

Graph Transformations

Analyzing transformations from f(x) to g(x) involves understanding shifts (up/down), stretches (steeper/flatter), and reflections (across axes).
Various combinations of these transformations can result in g(x) from f(x).

Explore the concepts of linear functions, transformations, and reflections in this quiz. Understand how to visualize transformations and how different functions relate to one another. Dive into examples of parallel lines, reflections, and shifts in graphs.