Mathematics — first biennium · Lesson
The form y=a(x−h)²+k makes the vertex, translations and concavity immediately visible.
Prerequisites: Functions and square of a binomial
Goal: Move from y=x² to y=a(x−h)²+k by reading vertex, translations and concavity.
In this lesson: Move from y=x² to y=a(x−h)²+k by reading vertex, translations and concavity.
New concept: A parabola has a vertex, the point where its direction of change turns. In y=a(x−h)²+k the vertex is (h,k); a>0 gives upward concavity and a minimum, while a<0 gives downward concavity and a maximum.
How to proceed: Start from y=x²: the sign of a determines concavity; h and k describe the shifts.
You can already assume: Functions and square of a binomial.
Move a, h and k to start from y = x² and immediately see concavity, translation and vertex position.
With a = 0 the curve loses this family's characteristic shape; also try a non-zero value.
The highlighted vertex makes translation, axis of symmetry and concavity visible.
Completed exercises: 0/5 · Correct on first attempt: 0
Start from y=x²: the sign of a determines concavity; h and k describe the shifts.
Data / situation: y=−(x−3)²−3
ConceptStep separates what is required from what is recommended or useful as reinforcement. Only required prerequisites block your progress.
Relationships: solid line = required · dashed line = recommended · dotted line = reinforcement.