Vertex form is y = a(x − h)² + k, and the vertex is (h, k). The form subtracts h, so in (x − 3)² the vertex is at x = 3. It describes y = x² shifted h right, k up, and stretched by a. Completing the square converts standard form to vertex form.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Watch the sign of h
y = (x + 2)² − 1 is (x − (−2))² − 1: vertex (−2, −1). Set the bracket to zero to find the vertex x: x + 2 = 0, x = −2. It works every time and avoids sign confusion.
As a transformation
y = 2(x − 3)² − 4: start from y = x², shift right 3, stretch vertically by 2, shift down 4. Inside acts horizontally and backwards; outside acts vertically as written.
Completing the square with a coefficient
y = 2x² − 12x + 5: factor the 2 from the x terms, y = 2(x² − 6x) + 5. Add 9 inside, which is 18 in total, so subtract 18: y = 2(x − 3)² + 5 − 18 = 2(x − 3)² − 13.
Back to standard form
y = (x − 3)² − 4 = x² − 6x + 9 − 4 = x² − 6x + 5. Expanding returns the original, which is the check on any completed square.
Why bother
The vertex is the maximum or minimum. y = (x − 3)² − 4 has minimum −4 at x = 3, read off directly. Vertex form answers most optimisation questions in one line.
Writing from the vertex
Vertex (1, 5), through (3, −3): y = a(x − 1)² + 5, and −3 = a(4) + 5, so a = −2. y = −2(x − 1)² + 5. The vertex gives h and k; one more point gives a.
Step 2: Try It Yourself
Tap and try it out.
Convert y = 2x² − 12x + 5 to vertex form, describe it as a transformation of y = x², and state its minimum
- 1y = 2(x² − 6x) + 5factor 2 from the x terms
Step 0 of 4
Step 3: In Real Life
A satellite dish
A dish is y = x² shifted and stretched, and vertex form gives its focus at a glance. Engineers write dishes in vertex form so the receiver can be placed.
Step 4: Watch an Example
One step at a time.
Watch Kofi Complete the Square
Kofi converts y = x² − 6x + 5 to vertex form.
- Step 1
Half of −6 is −3, and squaring gives 9. He writes y = (x² − 6x + 9) + 5 − 9, adding and subtracting the same 9.
Step 5: Your Turn
Practice makes it stick.
The Vertex
Problem 1 of 2
y = (x − 3)² − 4. What is the x-coordinate of the vertex?
The Minimum
Problem 2 of 2
Same equation. What is the minimum value of y?
Vertex Form
1 of 8
y = (x − 5)² + 2. Vertex x-coordinate?
2 of 8
y = (x + 4)² − 1. Vertex x-coordinate?
3 of 8
y = (x − 2)² + 7. What is the minimum value?
4 of 8
To complete the square on x² − 8x, what is added?
5 of 8
To complete the square on x² + 10x, what is added?
6 of 8
y = −(x − 1)² + 9. What is the maximum value?
7 of 8
Put the completing-the-square steps in order.
- 1Square that result.
- 2Add and subtract it inside the expression.
- 3Write the perfect square and collect the constants.
- 4Halve the coefficient of x.
8 of 8
y = (x − 7)² − 3. Vertex y-coordinate?
Step 6: Quick Check
Show what you know.
Question 1 of 2
y = (x − 6)² + 1. What is the vertex x-coordinate?
Question 2 of 2
Why does y = (x − 3)² have its vertex at x = 3?
What You Learned
- Vertex form is y = a(x − h)² + k, with vertex (h, k); set the bracket to zero to find h.
- Completing the square converts standard form to vertex form; factor a out first if needed.
- Vertex form reads as a transformation of y = x² and gives the maximum or minimum in one line.
- Expand to check, and use one extra point to find a from a known vertex.