A graph transformation changes where each point on a function’s graph lands. A horizontal shift changes the input inside the function: shifting right uses subtraction because the new input must reach the old one. A reflection across the -axis negates outputs, so its minus sign goes outside. Don’t add a rightward shift to the output.
Hints
- Hint 1
A vertex is a parabola’s turning point. In , the smallest squared value is , when the inside equals . What input makes ?
- Hint 2
A reflection across the -axis sends each output to . So the minus sign goes outside the function. Which way will the reflected parabola open?
- Hint 3
For a shift right by , the new input must be greater than the old one. Replacing with feeds that old input into . Where will its vertex land?
Step-by-step
Transform the graph one move at a time
Step 1Locate the original vertex
Type in Desmos and click the vertex, where the parabola turns. Desmos shows : the square is when .
- Step 2
Reflect the outputs
Type on the next line. Desmos shows a curve that opens downward and still has vertex . Reflecting across the -axis negates each output, not each input.
- Step 3
Shift the reflected graph right
Type and click its vertex. Desmos shows . A shift right by uses inside the function: at the new input , the old function receives , its vertex input.
- Step 4
Write the new rule without
The entire is the input of . Replace with , using :
- Step 5
Simplify the inside
Combine the constants inside the square:
It opens downward and has its vertex at , matching both moves. Choice D.