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: Picture It
Start with something you already know.
Tiling the corner of a patio
- Picture a square patio, x feet along each side.
- Along one edge you lay a strip 3 feet wide, and along the next edge another.
- Patio and strips together almost make a bigger square, x + 3 feet on a side.
- One gap is left in the corner, a small square 3 feet by 3 feet.
- Nine square feet of tile fill that gap, and the big square is complete.
x² + 6x is the patio plus the two strips. Each strip is half of 6 wide, which is 3.
The missing corner is that half squared, 9, and x² + 6x + 9 is the whole square (x + 3)².
Completing the square adds that corner, then takes the same amount away, so y does not change.
With x² − 6x the strips are cut away instead; they overlap in a corner that is still 3 by 3.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
y = (x − 3)² + 1. Work out y at x = 1, 2, 3, 4 and 5. What is the smallest value of y you find?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Maya Complete the Square
Maya puts y = x² + 4x + 1 into vertex form, reads off the vertex, and checks her work two ways.
- Step 1
Maya looks at the x term, 4x. Half of 4 is 2, and 2 squared is 4. That 4 is the number that turns x² + 4x into a perfect square.
Watch Dev Handle a Number in Front
Dev puts y = 2x² + 8x + 3 into vertex form. He does the first two steps, and you finish it.
- Step 1
The 2 in front of x² has to come out first, but only from the two x terms: y = 2(x² + 4x) + 3. The 3 stays outside, untouched.
Watch Sam Write It From the Vertex
A parabola has its vertex at (2, 1) and passes through (4, 9). Sam writes its equation in vertex form.
- Step 1
The vertex fills in h and k at once: y = a(x − 2)² + 1. Only a is still unknown.
Step 6: Your Turn
Practice makes it stick.
The Top of the Toss
Problem 1 of 17
A ball is tossed upward. Its height in feet, t seconds later, is y = −16(t − 2)² + 70. What is the greatest height the ball reaches, in feet?
feet
The Cheapest Chair
Problem 2 of 17
A small workshop makes x chairs a day. The cost of each chair, in dollars, is C = x² − 10x + 40. What is the lowest cost per chair, in dollars?
dollars
Vertex Form
Set the bracket to zero for h. Halve, then square, to complete the square.
Problem 3 of 17
y = (x + 5)² − 2. What is the y-coordinate of its vertex?
Problem 4 of 17
Complete the square to write y = x² − 8x + 10 in vertex form. Type the right-hand side.
Problem 5 of 17
What number completes the square on x² + 14x?
Problem 6 of 17
y = 3(x − 1)² + 4. Which moves take y = x² to this parabola?
Problem 7 of 17
Write y = 2x² − 8x + 1 in vertex form. What is the y-coordinate of the vertex?
Whose Step?
One line in each page is a slip. Find it, then fix it.
Problem 8 of 17
Priya completed the square on y = x² + 10x + 4. One line is a slip. Which?
Priya's vertex form for y = x² + 10x + 4
- 1half of 10 is 5; squared, 25
- 2y = (x² + 10x + 25) + 4 + 25
- 3y = (x + 5)² + 29
- Answervertex (−5, 29)
Problem 9 of 17
Priya’s bracket (x + 5)² is right. What number belongs after it, so that the vertex form equals y = x² + 10x + 4?
Problem 10 of 17
Ravi completed the square on y = 3x² + 12x + 2. One line is a slip. Which?
Ravi's vertex form for y = 3x² + 12x + 2
- 1y = 3(x² + 4x) + 2
- 2y = 3(x² + 4x + 4) + 2 − 4
- Answery = 3(x + 2)² − 2
Problem 11 of 17
Write the vertex form Ravi should have reached for y = 3x² + 12x + 2. Type the right-hand side.
Problem 12 of 17
Leo checked y = (x − 4)² + 1 by expanding it back to standard form. One line is a slip. Which?
Leo's expansion of y = (x − 4)² + 1
- 1y = (x − 4)² + 1
- 2y = (x − 4)(x − 4) + 1
- 3y = x² + 16 + 1
- Answery = x² + 17
Problem 13 of 17
Expand y = (x − 4)² + 1 correctly. Type the right-hand side, with no brackets.
Pick the Move
Choose the best way in. No need to finish the problem.
Problem 14 of 17
Noor wants y = 4x² + 8x + 1 in vertex form. Which first move keeps her on track?
Problem 15 of 17
A parabola has its vertex at (3, 2) and passes through (5, 10). Which is the best start to its equation?
Same Parabola?
Judge each one by its vertex or by expanding. No graphing needed.
Problem 16 of 17
Select every equation that draws the same parabola as y = x² − 4x + 7.
Problem 17 of 17
Select every equation whose minimum value is −4.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
y = a(x − 2)² + k passes through (0, 5) and (1, 2). Type the right-hand side in vertex form.
Stretch 2 of 2
y = x² + bx + c has its vertex at (3, −2). What is c?
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.
For the grown-up
The misconception this lesson targets is treating completing the square as adding a number, rather than adding and taking away the same amount. It shows two ways: the number added inside is added again outside, so y changes; or, with a coefficient in front, only the inside amount is taken away when the bracket multiplies it. Its cousin is reading the sign of h forwards, so (x + 5)² puts the vertex at x = 5. Expanding back to standard form, or substituting the vertex x into the original, catches both in one line.