Grade by Grade
A. Characteristics of Functions · Quadratic Functions

Maximum and Minimum Values by Completing the Square

A2.2About 70 minutes
Learning goals
  • I can complete the square on f(x) = ax² + bx + c when a ≠ 1 and when half of b is a fraction.
  • I can state the maximum or minimum value of a quadratic function and where it occurs.
  • I can check a vertex form by expanding or by substitution.
  • I can find the maximum or minimum in a real-world context.

Words to know

completing the square
Rewriting ax² + bx + c as a(x − h)² + k by adding and subtracting the square of half the x-coefficient.
perfect-square trinomial
A trinomial that factors as a square, like x² + 5x + 25/4 = (x + 5/2)².
vertex form
f(x) = a(x − h)² + k; the vertex is (h, k).
optimal value
The maximum or minimum value of a function, k in vertex form.

Why complete the square again?

In Grade 10 you completed the square when half of b was a whole number, like x² + 6x. Real functions are rarely that friendly. A store's profit, a football's height or the area of a pen can have coefficients like 5, −4.9 or ½. In this lesson you complete the square with any coefficients, including fractions and decimals, so you can find the maximum or minimum value of any quadratic function given as f(x) = ax² + bx + c. That is exactly what curriculum expectation A2.2 asks for.

The method in four steps

To write f(x) = ax² + bx + c in vertex form f(x) = a(x − h)² + k:

  1. Factor a out of the x² and x terms only: f(x) = a(x² + (b/a)x) + c.
  2. Take half of the new x-coefficient and square it.
  3. Add and subtract that square inside the bracket. (Adding zero keeps the function the same.)
  4. Move the subtracted square out of the bracket, multiplying it by a, then combine constants.

The vertex is (h, k). If a > 0, k is the minimum value; if a < 0, k is the maximum value. Either one happens when x = h.

x²2.5x2.5x6.25x2.5x2.5x² + 5x: split the 5x intotwo strips of 2.5x each.The dashed corner is missing:2.5 × 2.5 = 6.25 = 25/4.x² + 5x + 25/4 = (x + 5/2)²
Completing the square for x² + 5x. Half of 5 is 2.5, so each side gets a 2.5x strip. The missing corner is 2.5² = 6.25, which is 25/4.
Worked example

Example 1: a whole-number a

Find the minimum value of f(x) = 3x² + 12x − 2.

  1. Factor 3 from the first two terms: f(x) = 3(x² + 4x) − 2.
  2. Half of 4 is 2, and 2² = 4. Add and subtract 4 inside: f(x) = 3(x² + 4x + 4 − 4) − 2.
  3. Move −4 out, multiplying by 3: f(x) = 3(x² + 4x + 4) − 12 − 2.
  4. f(x) = 3(x + 2)² − 14.
  5. a = 3 > 0, so the parabola opens up. The minimum value is −14, when x = −2.
  6. Check: f(−2) = 3(4) − 24 − 2 = 12 − 24 − 2 = −14 ✓.

Minimum value −14 at x = −2.

Worked example

Example 2: an odd b (fractions)

Write f(x) = x² + 5x + 1 in vertex form and state its minimum value.

  1. Half of 5 is 5/2, and (5/2)² = 25/4.
  2. f(x) = x² + 5x + 25/4 − 25/4 + 1.
  3. f(x) = (x + 5/2)² − 25/4 + 4/4 (write 1 as 4/4).
  4. f(x) = (x + 5/2)² − 21/4.
  5. Check with decimals: f(−2.5) = 6.25 − 12.5 + 1 = −5.25 = −21/4 ✓.

f(x) = (x + 5/2)² − 21/4. Minimum value −21/4 = −5.25 at x = −5/2.

Worked example

Example 3: a negative a and an odd b

Find the maximum value of f(x) = −2x² + 6x + 5.

  1. Factor −2 from the first two terms: f(x) = −2(x² − 3x) + 5. (Check the sign: −2 × (−3x) = +6x ✓.)
  2. Half of −3 is −3/2, and (−3/2)² = 9/4: f(x) = −2(x² − 3x + 9/4 − 9/4) + 5.
  3. Move −9/4 out, multiplying by −2: −2 × (−9/4) = +9/2. So f(x) = −2(x − 3/2)² + 9/2 + 5.
  4. f(x) = −2(x − 3/2)² + 19/2.
  5. a = −2 < 0, so the maximum value is 19/2 = 9.5, when x = 3/2 = 1.5.
  6. Check: f(1.5) = −2(2.25) + 9 + 5 = −4.5 + 14 = 9.5 ✓.

Maximum value 9.5 at x = 1.5.

−101234−4−2024681012xymaximum (1.5, 9.5)(0, 5)(3, 5)x = 1.5
The graph of f(x) = −2x² + 6x + 5. The points (0, 5) and (3, 5) are mirror images across the axis x = 1.5, and the maximum value 9.5 is on that axis.
Worked example

Example 4: a fractional a

Find the minimum value of f(x) = ½x² − 3x + 4.

  1. Factor ½ from the first two terms. Dividing −3x by ½ gives −6x: f(x) = ½(x² − 6x) + 4.
  2. Half of −6 is −3, and (−3)² = 9: f(x) = ½(x² − 6x + 9 − 9) + 4.
  3. Move −9 out, multiplying by ½: f(x) = ½(x − 3)² − 9/2 + 4.
  4. f(x) = ½(x − 3)² − ½.
  5. Check: f(3) = 4.5 − 9 + 4 = −0.5 ✓.

Minimum value −½ at x = 3.

Worked example

Example 5: in context

A football is kicked so that its height in metres after t seconds is h(t) = −4.9t² + 14.7t + 1.2. Find the maximum height and when it happens.

  1. Factor −4.9 from the first two terms: 14.7 ÷ (−4.9) = −3, so h(t) = −4.9(t² − 3t) + 1.2.
  2. Half of −3 is −1.5, and (−1.5)² = 2.25: h(t) = −4.9(t² − 3t + 2.25 − 2.25) + 1.2.
  3. Move −2.25 out: −4.9 × (−2.25) = +11.025. So h(t) = −4.9(t − 1.5)² + 11.025 + 1.2 = −4.9(t − 1.5)² + 12.225.
  4. a = −4.9 < 0, so the maximum is 12.225 m at t = 1.5 s.

The football reaches about 12.2 m after 1.5 s.

Watch the sign when the square leaves the bracket

The number you subtract inside the bracket is multiplied by a on its way out. If a is negative, a subtraction becomes an addition. Always check your vertex form by expanding it, or by substituting x = h into the original function.

Hands-on activity

Paper tiles with half strips

You need: paper, a ruler, a pencil and scissors (take care with scissors; a grown-up can help).

  1. Let x = 10 cm. Cut one 10 cm by 10 cm square (an x² tile) and five 10 cm by 1 cm strips (x tiles). This models x² + 5x.
  2. You want to place half of the strips along the right side of the square and half along the bottom. Five strips do not split evenly, so cut one strip lengthwise into two 10 cm by 0.5 cm strips. Now place 2.5 strips on each side.
  3. Measure the empty corner. It is a 2.5 cm by 2.5 cm square. Cut it out of paper. Its area is 6.25 cm², which is (5/2)².
  4. Write the equation your shape shows: x² + 5x + 25/4 = (x + 5/2)².
  5. Repeat with three strips (x² + 3x). What size is the missing corner?

What to look for: for x² + 3x, each side gets 1.5 strips and the corner is 1.5 cm by 1.5 cm, an area of 2.25 cm² = 9/4, so x² + 3x + 9/4 = (x + 3/2)².

Just for fun

Completing the square is the only time a math teacher will tell you that adding zero is a really big deal.

Your turn

1

Write f(x) = x² + 8x + 3 in vertex form and state the minimum value.

2

Write f(x) = x² − 3x + 2 in vertex form.

3

Find the minimum value of f(x) = 2x² − 10x + 3.

4

Find the maximum value of f(x) = −x² + 7x − 4.

5

Find the maximum value of f(x) = −3x² − 12x + 1.

6

Find the minimum value of f(x) = 0.5x² + 2x − 1.

7

A ball's height in metres is h(t) = −5t² + 30t + 2, where t is in seconds. Find its maximum height and when it occurs.

More practice: warm-up

1

What number completes the square for x² + 10x?

2

What number completes the square for x² − 7x?

3

What number completes the square for x² + x?

4

State the vertex and the minimum value of f(x) = (x − 4)² + 2.

5

State the maximum value of f(x) = −(x + 1)² + 6.

More practice: on your own

1

Find the minimum value of f(x) = 4x² + 4x − 3.

2

Find the maximum value of f(x) = −0.25x² + 2x + 5.

3

Write f(x) = 3x² − 5x + 1 in vertex form.

4

Jun wrote 2x² + 12x + 1 = 2(x + 3)² + 1 − 9 = 2(x + 3)² − 8. Find the mistake and correct it.

5

A skate-rental shop's daily revenue is R(x) = −20x² + 400x + 6000 dollars, where x is the number of $1 price increases. Find the maximum revenue and the number of increases.

Challenge

1

Complete the square on f(x) = ax² + bx + c to show that the vertex is at x = −b/(2a).

2

f(x) = 2x² + kx + 8 has a minimum value of 0. Find all possible values of k.

3

The minimum value of f(x) = x² − 6x + c is 5. Find c, and the minimum value of g(x) = 3f(x).

Check your understanding

Quiz

1.

What number must be added and subtracted to complete the square for x² − 12x?

2.

Which is the vertex form of f(x) = x² + 3x + 1?

3.

What is the maximum value of f(x) = −2(x − 1)² + 7?

4.

What is the minimum value of f(x) = 2x² − 8x + 1?

5.

To complete the square for f(x) = −3x² + 9x + 4, you first factor −3 from the first two terms. What is inside the bracket?

6.

A ball's height is h(t) = −5(t − 2)² + 23 metres after t seconds. Which statement is true?

For parents and teachers

Big idea: Expectation A2.2: find the maximum or minimum of f(x) = ax² + bx + c algebraically. This lesson extends Grade 10 completing the square to fractions and decimals.

How to help: Ask your teen to check every vertex form by substituting x = h into the original function. If the answer matches k, the work is right.

Common difficulties:

  • Forgetting to multiply the subtracted square by a when it leaves the bracket.
  • Sign errors when a is negative (factor −2 from 6x and you get −3x).
  • Fraction arithmetic, such as −25/4 + 1. Encourage writing 1 as 4/4.
  • Calling k a minimum when a < 0. The sign of a decides.