Unit 5, Lesson 7
Expressing Transformations of Functions Algebraically
Integrated Math 3
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You can use the equation of a function to write an equation for its transformation. For example, let . Take the graph of , reflect it across the -axis, translate it up 10 units, and translate it left 3 units. What is an equation for this new function? The new function is related to by , since
G of x is equal to negative f times the quantity of x plus three, plus 10. Arrow pointing to negative f with label “reflect across x axis.” Arrow pointing to x plus three with label “shift left 3.” Arrow pointing to plus 10 with label “shift up 10.”
Which means .
Sometimes you can recognize from the expression for a function that it is the transformation of a simpler function. For example, consider:
One way to obtain the expression for from is:
So the graph of is obtained from the graph of by translating left 5 units, reflecting across the -axis, and translating up 10 units. Consider the point on the graph of . After translating, reflecting, and translating again, it becomes the point on the graph of .