A transformation of a quadratic moves its vertex, the parabola’s turning point. Find the vertex’s new input by making the expression inside equal its old input. Then change its height using the operations outside , and test the proposed equations at that new input in Desmos. The plus sign inside moves the graph left, not right.
Hints
- Hint 1
The vertex is the parabola’s turning point. Shifting and flipping a parabola move that point, but it remains the turning point. Start with the marked vertex: what input and output does it give you?
- Hint 2
The old vertex uses input . To get that same output from , make the entire new input equal . What value of solves ?
- Hint 3
Once you’ve found the new input, change the old height . The multiplies the height before is added. Then test which proposed equation gives that new height at the new input.
Step-by-step
Track the vertex and test the equations
Step 1Find the vertex’s new input
The graph marks the vertex, or turning point, at , so . To get that output from , set the input inside equal to the old input:
Subtract :
The moves the vertex left, from input to input .
- Step 2
Find the vertex’s new height
At the new input, still outputs . Type into Desmos; it prints . The changes the old height before the is added, so the new vertex is .
- Step 3
Test the choices at the new vertex
Type , then type each choice’s right-hand side with replaced by . Desmos prints , , , and in choice order. Only gives the required height at . Its positive coefficient also fits the upward-opening graph produced when flips the original downward-opening parabola. So the transformed graph is . Choice A.