An axis of symmetry is the vertical line through a parabola’s turning point. If one point has output , mirror its input across the axis to find the other root. Write the quadratic in factored form, use a second point to find its multiplier in Desmos, then evaluate at the axis. A point with nonzero output is not a root.
Hints
- Hint 1
A root is an input that gives output . The axis of symmetry acts like a mirror for roots. If one root is one unit left of the axis, where is its matching root?
- Hint 2
Factored form has a factor for each root . Once you have both factors, use the point with nonzero output to find the multiplier in front.
- Hint 3
The vertex is the parabola’s turning point, so it lies on the axis of symmetry. After finding the multiplier, which input should you use to get the vertex’s -coordinate?
Step-by-step
Mirror the root, then evaluate
Step 1Find the matching root
A root is an input with output , so gives the root . The axis splits the parabola into mirror halves. Since is one unit left of , its matching root is one unit right, at . Mirror a root across the given axis to find its partner.
- Step 2
Write the function from its roots
Each root makes one factor zero. The roots and give the factored form . The multiplier is still unknown because roots alone don’t determine the parabola’s height.
- Step 3
Use the other point
The point says that input gives output . Put into the function: . Don’t treat as a root: its output is , not .
- Step 4
Find the multiplier in Desmos
Type . The tells Desmos to find the unknown multiplier. Under PARAMETERS, it shows .
- Step 5
Find the vertex’s height
The vertex lies on , so its height is . Type below the regression. Desmos uses its stored and prints ; the fraction button on this evaluated line shows . That’s the vertex’s -coordinate. Choice D.