Short Answer 
To understand function transformations, first identify the parent function (e.g., linear or quadratic). Next, determine the modifications applied, like vertical or horizontal shifts, stretching, or reflections, and finally rewrite the function incorporating these transformations, analyzing it in comparison to the parent function for shifts, stretch effects, and other attributes.
Step 1: Identify the Parent Function
The first crucial step in understanding transformations is to establish the parent function. The parent function is the most basic form of a function without any transformations applied. Examples of common parent functions include linear functions (y = x), quadratic functions (y = x²), and trigonometric functions (y = sin(x)). Recognizing the parent function allows you to comprehend what modifications have been made to it.
Step 2: Determine the Changes Applied
Once the parent function is identified, the next step is to examine any modifications that act upon it. This can include various transformations such as:
- Vertical shifts (upward or downward)
- Horizontal shifts (leftward or rightward)
- Stretching or compressing vertically
- Reflections across axes
By pinpointing these changes, you can better understand how the function’s outputs have been affected relative to the parent function.
Step 3: Rewrite and Analyze the Function
The final step is to rewrite the function incorporating the transformations identified. This involves adjusting the function’s elements such as coefficients and variables according to the changes observed. After rewriting, analyze the new function by comparing it to the parent function, focusing on:
- Direction and amount of shifts
- Narrowness or wideness due to stretching/compression
- Axis of reflection and its effect
- Changes in amplitude or period for periodic functions
This analysis will give a comprehensive view of how the graph of the modified function relates to its parent function.