Print this Page
4.6 – Transformation of Parent Functions
Objectives
 Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from a horizontal or vertical shift to the parent function.
 Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from a vertical stretch or compression to the parent function.
 Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from flipping the parent function’s graph over the xaxis.
 Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from a combination of shifts, stretches, or compressions to the parent function.
Key Terms
 Transformation – A function that has been changed by shifting, stretching, compressing or flipping the original function.
Notes
Transformations of Functions 
 Shift: to MOVE a function vertically (up or down) or horizontally (left or right)
 Up / Down: Add or subtract a value to the end of the equation
 Left / Right: Subtract a positive or negative value from the xterm
 Stretch: to change the SHAPE of a function (make it tall and skinny)
 Multiply the xterm by a number greater than 1
 Compress: to change the SHAPE of a function (to make it short and wide)
 Multiply the xterm by a fraction or decimal between 0 and 1
 Flip (over an axis)
 Multiply the xterm by a negative

Examples 
 Ex. Shifting Linear Functions
 Notice how shifting UP by 5 is the same as shifting LEFT by 5.
 Up (add 5 to the end of the equation): F(x) = x becomes G(x) = x + 5
 Left (subtract negative 5): F(x) = x becomes G(x) = (x – (5)), simplified: G(x) = x + 5
 Notice how shifting DOWN by 6 is the same as shifting RIGHT by 6
 Down (subtract 5 from the end of the equation): F(x) = x becomes G(x) = x – 6
 Right (subtract positive 6 from x): F(x) = x becomes G(x) = (x – 6), simplified: G(x) = x – 6
 Ex. Stretching and Compressing Linear Functions
 Notice how stretching by 4 is the same as increasing the steepness of the slope
 Stretching means multiplying the x by a number greater than 1
 The coefficient of x > 1
 Notice how compressing by is the same as decreasing the steepness of the slope
 Compressing means multiplying the x by a number between 0 and 1
 The coefficient of x: 0 < x < 1
 Ex. Multiple Transformations on One Graph
 Ex. Shifting Quadratic Functions
 Ex. Stretching and Compressing Quadratic Functions
 Ex. Multiple Transformations on One Graph
 Ex. Shifting Absolute Value Functions
 Shifting Vertically: Up and Down: add or subtract the value from the end of the equation
 Shifting Horizontally: Left and Right: subtract a positive (right) or negative (left) value from the x
 Ex. Stretching and Compressing Absolute Value Functions
 Ex. Multiple Transformations on One Graph
 Ex. Shifting Reciprocal Functions Vertically
 Ex. Shifting Reciprocal Functions Horizontally
 Notice that the xvalue (in the denominator) is affected
 Ex. Stretching and Compressing Reciprocal Functions
 As expected, we multiply the xterm by the stretched or compressed value, then simplify
 Ex. Multiple Transformations on One Graph

Important!
Practice (Apex Study 4.6)
Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=2452