Same Math › Algebra 1 · grades 8–10 › Vertex form and shifts
What is vertex form?
Vertex form, y = a(x − h)² + k, shows the vertex right away: it's (h, k). Kids read it as moves from y = x²: h shifts it left or right, k shifts it up or down, and a flips it or makes it narrower or wider. The tricky part is the sign of h: (x − 2)² moves right 2.
- a opens it up (+) or down (−) and stretches it.
- h moves it left or right, with the opposite sign.
- k moves it up or down. The vertex is (h, k).
Why schools teach it
Seeing a graph as a moved copy of y = x²
Students learn how changing an equation moves its graph (HSF.BF.B.3) and rewrite quadratics to reveal the vertex (HSF.IF.C.8a). Vertex form is where both meet.
It also connects to completing the square, which turns standard form into vertex form, and sets up the same shifts for every function in Algebra 2.
At the kitchen table
What to say, and what to skip
Say this
What's the vertex?
Read h and k. Watch the sign of h.Which way does it move from y = x²?
Does it open up or down?
Is it narrower or wider than y = x²?
Compare |a| to 1.
Skip this
(x − 3)² moves left 3.
It moves right 3. x = 3 is where (x − 3) becomes zero.- Reading the vertex of y = (x + 4)² − 1 as (4, −1).It's (−4, −1).
- Thinking a bigger a makes it wider.Bigger |a| makes it narrower.
See it in action
Describing y = −2(x + 1)² + 3
Vertex
(−1, 3)Shifts from y = x²
left 1, up 3a = −2
opens down, narrower than y = x²
The vertex is the highest point here, because it opens down.
Free practice sheet
Make a vertex form sheet
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Parents also ask
Why does the sign of h flip?
The vertex is where x − h = 0, so x = h. For (x + 4), that's x = −4.
How do you get vertex form from standard form?
Complete the square: y = x² + 6x + 5 = (x + 3)² − 4, so the vertex is (−3, −4).
What's the axis of symmetry in vertex form?
x = h. It goes straight through the vertex.
Is there a factored form too?
Yes: y = a(x − p)(x − q) shows the x-intercepts p and q. Algebra 1 uses all three forms.