Translations of Functions - Algebra-2

1. Fundamental Concepts

Translation of a Function: The movement of a function’s graph along the horizontal (x-axis) or vertical (y-axis) direction in the coordinate plane. Translations do not change the shape, size, or orientation of the function; they only alter its position. Translations are one of the most basic types of function transformations, typically applied to a parent function.
Horizontal Translation: The shift of a function’s graph along the x-axis (left or right), caused by adding or subtracting a value to/from the independent variable x.
Vertical Translation: The shift of a function’s graph along the y-axis (up or down), caused by adding or subtracting a value to/from the entire function (dependent variable).

Let the parent function be . The forms and rules of translated functions are as follows:

Translation Type	Direction and Distance	Core Rules Summary
Horizontal Translation	Shift right by h units ()	"Subtracting" from x shifts right; "adding" to x shifts left (opposite to the sign: = right, = left).
Horizontal Translation	Shift left by h units ()
Vertical Translation	Shift up by k units ()	"Adding" to the entire function shifts up; "subtracting" from the entire function shifts down (consistent with the sign: = up, = down).
Vertical Translation	Shift down by k units ()
Combined Translation	First horizontal shift by h, then vertical shift by k	The order of translations does not affect the result; they can be performed step-by-step or combined directly.

Question: Given the parent function , write the expression of the function after it is shifted 3 units to the right. Answer: By the rule of horizontal translation, shifting right by 3 units requires subtracting 3 from x, so the result is .
Question: How is the function obtained by translating the parent function ? Answer: Adding 2 to the entire parent function means shifting it 2 units upward.

Question: The parent function is first shifted 1 unit to the left, then 4 units downward. Find the expression of the transformed function and state the coordinates of its vertex (starting point). Answer:
- Shifting left by 1 unit: ;
- Shifting downward by 4 units: ;
- The original starting point of the parent function is ; after translation, the new starting point is .
Question: For the function , explain how it is obtained by translating the quadratic parent function , and determine its vertex coordinates. Answer:
- The transformation indicates a shift 2 units to the right;
- Adding 5 to the entire function indicates a shift 5 units upward;
- The vertex coordinates are (derived from shifting the original vertex ).

Question: A quadratic function’s graph is obtained by shifting right by h units and then up by k units. The graph passes through the points and , and its vertex lies on the line . Find the values of h and k. Answer:
- The translated function is , with vertex ;
- Since the points and have the same y-coordinate, the axis of symmetry is , so ;
- The vertex lies on ; substituting gives ;
- Thus, and .

Determine translation direction and distance:
- Horizontal translation: Check the "addition/subtraction" applied to x: = shift right by h; = shift left by h ().
- Vertical translation: Check the "addition/subtraction" applied to the entire function: = shift up by k; = shift down by k ().
Track key points during translation: Key points of the parent function (e.g., vertex, intercepts with axes, asymptotes) serve as "markers" for translation. For example:
- The vertex of the quadratic function becomes after translation to ;
Reverse derivation: Find parent function transformations from the translated function: If the translated function is known, reverse the operations to recover the parent function. For example: is derived by shifting the parent function left 1 unit and down 2 units; reversing these steps (shift right 1 unit, up 2 units) returns to .
Verify with graphs: Plot the parent function and the translated function, then compare the positions of key points to confirm the translation (shape remains unchanged, only position differs).
Handle combined translations step-by-step: For translations involving both horizontal and vertical shifts, first process the horizontal shift (modifying x), then the vertical shift (modifying the entire function). Step-by-step calculation reduces errors.