Vertex Form
y = a(x − h)² + k shows the vertex (h, k) directly.
About this topic
Vertex form y = a(x − h)² + k shows the vertex (h, k) directly. The value a controls opening direction and width, h shifts the parabola left or right (note the sign is opposite to what you see), and k shifts it up or down.
This form makes graphing and finding the maximum or minimum quick. For y = 2(x − 3)² + 1, the vertex is (3, 1), the parabola opens up, and it is narrower than y = x². For y = −(x + 2)² + 5, the vertex is (−2, 5) and it opens down with a maximum value of 5. To write the equation from a vertex and another point, substitute the vertex, plug in the point, and solve for a.
Vertex form is a transformation view: start with the basic parabola y = x² and slide, stretch and flip it. Designers use this to fit curves to arches, and athletes' trajectory models read the peak height and its position straight from h and k.
The sign trap is the biggest issue: (x + 2) means h = −2. Also remember that a multiplies the squared part only, not the k. Completing the square converts standard form into vertex form.
Key ideas
- h shifts left or right
- k shifts up or down
- a stretches or flips
Quick check
Use cases: where Vertex Form helps
- Homework: review the key idea behind Vertex Form before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Vertex Form without notes.
- Visual learning: play with the interactive lab to see how Vertex Form changes when you adjust its inputs.
- Teaching a friend: explain Vertex Form in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Vertex Form?
y = a(x − h)² + k shows the vertex (h, k) directly.
What will I learn in Vertex Form?
h shifts left or right. k shifts up or down. a stretches or flips.
Is Vertex Form part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Vertex Form lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Vertex Form?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.