# 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 x-axis.
• 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 x-term
• Stretch: to change the SHAPE of a function (make it tall and skinny)
• Multiply the x-term by a number greater than 1
• Compress: to change the SHAPE of a function (to make it short and wide)
• Multiply the x-term by a fraction or decimal between 0 and 1
• Flip (over an axis)
• Multiply the x-term 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 $\frac{1}{2}$ 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. 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 x-value (in the denominator) is affected

• Ex. Stretching and Compressing Reciprocal Functions
• As expected, we multiply the x-term by the stretched or compressed value, then simplify

• Ex. Multiple Transformations on One Graph