Grade by Grade
Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Zeros and Maximum or Minimum Values

4.1 4.2About 70 minutes
Learning goals
  • I can determine the zeros and the maximum or minimum value of a quadratic relation from its graph.
  • I can determine the zeros and the maximum or minimum value from an equation in any form.
  • I can use x = −b ÷ (2a) or partial factoring to find the axis of symmetry from standard form.
  • I can solve simple optimization problems, such as the largest product or the largest area.

Words to know

maximum value
The greatest value of y; the y-coordinate of the vertex of a parabola that opens down.
minimum value
The least value of y; the y-coordinate of the vertex of a parabola that opens up.
optimization
Finding the best value (largest or smallest) of a quantity, such as the greatest area or the least cost.
zeros
The x-values where y = 0; the x-intercepts.

The highest point

On Canada Day, a firework shell rises, slows down, stops for an instant at its highest point, and then falls. A company that sells hoodies finds that if the price is too low it makes little money, and if the price is too high nobody buys, so some price in the middle gives the most revenue. A farmer with a fixed length of fence wants the largest possible pen.

In all three situations, a quadratic relation has a vertex, and the y-coordinate of the vertex is the answer to the question "What is the biggest (or smallest) possible value?" This lesson collects every way you know to find that vertex, and the zeros, from a graph or from an equation.

Max or min?

  • If a > 0, the parabola opens up: the vertex is the lowest point, and its y-coordinate is the minimum value.
  • If a < 0, it opens down: the vertex is the highest point, and its y-coordinate is the maximum value.
  • The x-coordinate of the vertex tells you where (or when) the max or min happens.

From a graph

On a graph, the maximum or minimum is the vertex and the zeros are the x-intercepts. With graphing technology, you can tap or trace to these points to get accurate decimal values. With a printed graph, read them as carefully as the scale allows.

−10123456789−4−20246810xy(0, 0)(8, 0)maximum (4, 8)
A parabola that opens down. Read from the graph: zeros at x = 0 and x = 8, a maximum value of 8 when x = 4. Its equation is y = −0.5(x − 4)² + 8.

From an equation: four routes to the vertex

  1. Vertex form y = a(x − h)² + k: the max or min is k, at x = h.
  2. Factored form y = a(x − r)(x − s): the vertex is halfway between the zeros, at x = (r + s) ÷ 2. Substitute to get the max or min.
  3. Standard form: complete the square to get vertex form.
  4. Standard form: use partial factoring to find two points with the same y-value, then the axis halfway between them.

To find the zeros from an equation, solve y = 0 by factoring, by square roots from vertex form, or with the quadratic formula.

Vertex formread kFactored formaverage the zerosComplete the squareget vertex formPartial factoringfind the axis
Four algebraic routes to the maximum or minimum. They all find the same vertex; choose the one that fits the form you are given.
Worked example

Example 1: From factored form

Find the zeros and the maximum value of y = −2(x − 1)(x − 7).

  1. Zeros: x = 1 and x = 7.
  2. Axis: x = (1 + 7) ÷ 2 = 4.
  3. y = −2(4 − 1)(4 − 7) = −2(3)(−3) = 18.
  4. a = −2 < 0, so this is a maximum.

Zeros 1 and 7; maximum value 18 when x = 4.

Worked example

Example 2: Complete the square

Find the maximum value and the zeros of y = −3x² + 18x − 20.

  1. y = −3(x² − 6x) − 20 = −3(x² − 6x + 9 − 9) − 20 = −3(x − 3)² + 27 − 20.
  2. y = −3(x − 3)² + 7. Maximum value 7, at x = 3.
  3. Zeros: set y = 0: 3(x − 3)² = 7, so (x − 3)² = 7/3 and x = 3 ± √(7/3).
  4. x ≈ 3 − 1.528 = 1.47 or x ≈ 3 + 1.528 = 4.53. (The quadratic formula gives the same values.)

Maximum 7 at x = 3; zeros x ≈ 1.47 and x ≈ 4.53.

Worked example

Example 3: Partial factoring

Find the minimum value of y = 2x² + 12x + 5.

  1. Partial factor: y = 2x(x + 6) + 5.
  2. y = 5 when x = 0 and when x = −6, so the axis is x = −3.
  3. y = 2(9) + 12(−3) + 5 = 18 − 36 + 5 = −13.
  4. a = 2 > 0, so it is a minimum.

Minimum value −13 when x = −3.

Worked example

Example 4: The biggest product

Two numbers have a sum of 20. What is the largest possible product?

  1. Let one number be x. The other is 20 − x.
  2. Product: P = x(20 − x) = −x² + 20x.
  3. This is a parabola opening down (a = −1), with zeros x = 0 and x = 20.
  4. Axis: x = 10. P = 10(10) = 100.
  5. The numbers are 10 and 10.

The largest product is 100, when both numbers are 10.

024681012141618202224020406080100120width (m)area (m²)(10, 100)(4, 64)(16, 64)
Example 4 as a fence problem: 40 m of fence makes a rectangle with width x and length 20 − x, so the area is A = x(20 − x). The maximum area, 100 m², comes from a 10 m by 10 m square.

A shortcut worth knowing

Partial factoring y = x(ax + b) + c shows that the axis of symmetry of y = ax² + bx + c is always x = −b ÷ (2a). For y = 2x² + 12x + 5: x = −12 ÷ 4 = −3 ✓. Substitute this x-value to get the max or min. You will use this often in Grade 11.

Hands-on activity

String rectangles

You need: a piece of string or yarn exactly 40 cm long (tie the ends to make a loop), a ruler, four pins or four small objects such as erasers, and paper.

  1. Stretch the loop around the four objects to make a rectangle that is 4 cm wide. Measure the length. Record the width, length and area in a table.
  2. Repeat with widths of 6, 8, 10, 12, 14 and 16 cm.
  3. What do you notice about width + length each time?
  4. Which rectangle has the largest area? Plot area against width on graph paper.
  5. Write the area as an expression in w, and find its maximum algebraically.

Model answer: width + length is always 20 cm. Areas: 4 × 16 = 64, 6 × 14 = 84, 8 × 12 = 96, 10 × 10 = 100, 12 × 8 = 96, 14 × 6 = 84, 16 × 4 = 64 cm². The graph is a parabola opening down. A = w(20 − w), with zeros 0 and 20, axis w = 10, and maximum 100 cm² for a square.

Just for fun

The vertex of a downward parabola is the only point where a graph can honestly say: "It's all downhill from here."

Your turn

1

Find the minimum value of y = (x − 2)(x − 8) and where it occurs.

2

State the minimum value of y = 4(x + 1)² − 3 and where it occurs.

3

Find the maximum value and the zeros of y = −x² + 10x − 16.

4

Find the minimum value of y = 2x² − 8x + 1.

5

Find the zeros and the minimum value of y = x² + 6x.

6

Two numbers differ by 8. What is the smallest possible product, and what are the numbers?

7

Use the first graph in this lesson to state the zeros and the maximum value. Then check with the equation y = −0.5(x − 4)² + 8.

More practice: warm-up

1

State the minimum value of y = (x − 3)² + 2.

2

State the maximum value of y = −5(x + 4)² − 1.

3

At what x-value is the vertex of y = 3(x − 2)(x − 6)?

4

Does y = −2x² + 3x + 1 have a maximum or a minimum?

5

Find the minimum value of y = x(x − 10).

More practice: on your own

1

Find the vertex of y = −4x² + 24x + 3 using x = −b ÷ (2a). Is it a max or min?

2

Find the minimum value of y = 0.5x² − 4x + 3.

3

Two numbers have a sum of 10. What is the smallest possible value of the sum of their squares?

4

Find the zeros and the minimum value of y = x² − 8x + 12.

5

A farmer has 60 m of fence to make a rectangular pen. Find the dimensions that give the largest area, and the area.

Challenge

1

A parabola has zeros −2 and 6 and a minimum value of −32. Find its equation.

2

Show that y = x² − 2kx + k² + 3 has a minimum value of 3 for every value of k.

3

A farmer uses 60 m of fence for three sides of a rectangular pen; a long barn wall forms the fourth side. Find the largest possible area.

Check your understanding

Quiz

1.

What is the maximum value of y = −2(x − 5)² + 7?

2.

What is the minimum value of y = (x − 4)(x + 2)?

3.

What is the minimum value of y = x² − 8x + 20?

4.

Which relation has a maximum value?

5.

Two numbers have a sum of 14. What is their largest possible product?

6.

What is the vertex of y = 2x² − 12x + 7?

For parents and teachers

Big idea: Expectation 4.1 asks students to determine the zeros and the maximum or minimum value of a quadratic relation from its graph (using technology) or from its defining equation (using algebra). This lesson pulls together vertex form, factored form, completing the square and partial factoring, and starts applying them to optimization problems that the next two lessons develop further.

How to help: The string activity takes ten minutes and gives a memorable answer: for a fixed perimeter, the square has the largest area. Ask your teen to explain which route to the vertex they used and why.

Common difficulties:

  • Giving the x-coordinate of the vertex when the question asks for the max or min value (the y-coordinate), or the reverse.
  • Saying "maximum" for a parabola that opens up.
  • Arithmetic slips when substituting the axis value; checking with technology helps.