Functions, graphs and inverses practice question 3
The graph of is shifted 3 units to the right and then reflected in the -axis. Which formula describes the resulting graph?
Handle the two transformations separately, in the order they are described.
A horizontal shift changes the input: after moving 3 units right, the new graph at must have the height the old one had at . A reflection in the -axis changes the sign of the output.
Theory UBC Math 100 §1.5CLP-1 §0.4Key resultsAssumed from precalculus
Worked solution
Strategy
Apply one transformation at a time. Horizontal changes happen inside the function and run opposite to intuition; vertical changes happen outside and behave as written.
Working
Shifting 3 units right replaces x with x − 3. To reach the same height, the input must now be 3 larger.
Reflecting in the x-axis negates every output, so a minus sign goes outside the function.
Answer
The answer is y = −f(x − 3).
Check
With f(x) = x², the vertex should move from (0, 0) to (3, 0) and the parabola turn upside down, and −(x − 3)² does both.
Where this goes wrong
Writing y = −f(x + 3) because “right” feels like it should add. Inside the function the sign is reversed: x − 3 moves the graph right, x + 3 moves it left.
The idea behind it
The functions, graphs and inverses lesson, a figure to move, in a few steps
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