A function transformation sends a point on an old graph to a point on a new one. Read the point as an input-output pair. Match the input inside before changing the output outside . Use a Desmos regression to find the new input, then apply the outside changes in order. Inside subtraction moves a point right, not left.
Hints
- Hint 1
For a point on , the horizontal coordinate is the input and the vertical coordinate is the output. Read both from point , rather than from the parabola’s highest point.
- Hint 2
The expression inside must equal the old input. Set equal to point ’s horizontal coordinate to find where that point lands on the new graph.
- Hint 3
The minus sign outside makes a reflection across the -axis: it changes the old output’s sign. Then the moves that height up. Apply those changes in that order.
Step-by-step
Map point A through the transformation
Step 1Read the input and output at A
Trace point down to and across to . A point on means the input gives the output , so . The parabola’s highest point isn’t the point being moved.
- Step 2
Match the input inside f
To get the same output on the new graph, the expression inside must equal the old input , so . Don’t subtract from ’s coordinate: is the new coordinate you need to find.
- Step 3
Find the new horizontal coordinate
Type in Desmos. Use for the unknown coordinate and to ask Desmos to fit the equation. Under PARAMETERS, it shows , so moves to .
- Step 4
Reflect the old output
The minus sign outside changes the sign of the old output. Reflect across the -axis:
- Step 5
Shift up and name the point
The outside raises that reflected height by . Add it:
So the corresponding point is . Choice C.