The turning point of a quadratic is its highest or lowest point. In a word problem, its y-value is the best or worst result, and its x-value is the condition that gives it.
This lesson is part of SPM Mathematics quadratic functions and equations. It builds on finding roots from a quadratic graph, because the turning point sits halfway between the roots.
Worked example 1: a fenced plot
A farmer uses 40 m of fencing for a rectangular plot fenced on all four sides. One side is x metres long.
The other side is (20 − x) metres, because one length and one width together use half the fence, 20 m. The area is A = x(20 − x) = 20x − x².
Find the turning point. The area is zero when x = 0 or x = 20, so the roots are 0 and 20. The turning point has x = (0 + 20) ÷ 2 = 10.
Substitute: A = 10(20 − 10) = 100. Since the coefficient of x² is −1, the graph is ∩ and this is a maximum.
Write the answer. The maximum area is 100 m², when the plot is 10 m by 10 m. That means a square gives the greatest area for this fencing.
Worked example 2: unit cost
A workshop models the cost per item as C = x² − 10x + 40, in RM, where x is the number of hundreds of items made in a batch. The coefficient of x² is positive, so the graph is ∪ and the turning point is a minimum.
This one does not factorise into whole numbers, so use the axis of symmetry. For ax² + bx + c, the axis is x = −b ÷ 2a, here x = 10 ÷ 2 = 5.
Substitute: C = 25 − 50 + 40 = 15. The minimum cost is RM15 per item, when 500 items (x = 5 hundred) are made.
The mistake that costs marks
The common slip is giving the wrong coordinate. Asked for the maximum area, a student answers “10” because that is the x-value found first. The maximum area is the y-value, 100.
The second slip is dropping units and context. “Minimum = 15” earns less than “the minimum cost is RM15 per item”.
| Question asks | Use | Example 1 answer |
|---|---|---|
| Maximum or minimum value | y-value of turning point | 100 m² |
| Value of x that gives it | x-value of turning point | 10 m |
| Shape of the graph | Sign of the x² coefficient | ∩ so maximum |
A sentence template that works
Write: “The [maximum or minimum] [quantity] is [value and unit] when [variable meaning] is [value and unit].” Fill it from the problem’s own words.
For example 2: “The minimum cost per item is RM15 when 500 items are made.” The sentence shows you have both coordinates and know what each one means.
Check yourself
A ball’s height is h = 6t − t², where h is in metres and t is in seconds. Find the maximum height and when it occurs. Write your answer as a sentence.
Answer
The height is zero when t(6 − t) = 0, so at t = 0 and t = 6. The turning point has t = (0 + 6) ÷ 2 = 3.
h = 6(3) − 3² = 18 − 9 = 9.
The coefficient of t² is −1, so this is a maximum. The maximum height is 9 m, reached after 3 seconds.
What to study next
Practise choosing between roots, turning points and intercepts in the quadratic practice set. The quadratic graph and roots explorer lets you vary the numbers and watch the turning point move.
If you want a teacher to go through your word-problem wording, see online one-to-one Mathematics tuition.