A function transformation changes a graph by changing its inputs or outputs. A reflection across the -axis reverses the inputs; a translation up raises the outputs. You can use a sample value of the constant in Desmos to see the moves, then write a rule that works for every value. Keep the upward shift outside the function.
Hints
- Hint 1
A reflection across the -axis sends a point at to without changing its height. What should replace inside to reverse every input?
- Hint 2
A translation up changes each output, not the input. After reflecting the graph, where should you add so the whole graph moves upward?
Step-by-step
Reflect the input, then raise the output
Step 1Reflect the graph across the -axis
Set only to view the moves in Desmos; the final equation must still use . Type , the given , and . A reflection across the -axis sends to , so replacing the input with makes Desmos show the mirror image. To reflect across the -axis, replace with inside the function.
- Step 2
Move the reflected graph up
Type . Desmos shows 8 units above . An upward translation adds to every output, so the goes outside the function, not inside its denominator. Together, the moves give .
- Step 3
Write the rule for every value of
Put into the given function rule, then add outside it:
Combine the constants:
The input is reversed and each output is raised by , so this equation works for every , not only the value used in Desmos. Choice C.