Ch 2 · 2 Quadratic Functions · O Level Additional Mathematics 4037
Topic 2 of 6 in Algebra and Functions — O Level Additional Mathematics 4037 — 58 lessons.
Introduction
The path of a basketball, the area of a rectangle with a fixed perimeter, a company's profit as the price changes: each is a quadratic. Its graph is a parabola with one turning point, and finding that point (the maximum or minimum) is the key to most questions. (Syllabus topic 2.)
Key ideas — Maximum and minimum by completing the square
ax^2 + bx + c = a(x + p)^2 + q
🧑🏫 Teacher's explanation
Imagine you are throwing a ball into the air. The path it follows is a parabola. At the very peak of its flight, it stops moving upwards and starts moving downwards; this 'turning point' is the maximum height. In algebra, we can find this exact peak by rewriting our quadratic equation into a special form called 'vertex form'. Instead of looking at the equation as a messy sum of terms, we look at it as a single squared term that has been shifted around.
📖 Definitions
🔍 Worked example — Easy
Find the minimum value of f(x) = x^2 - 6x + 10 by completing the square.
🔍 Worked example — Standard
Express y = 3x^2 + 12x + 7 in the form a(x + p)^2 + q and state the nature of the turning point.
🔍 Worked example — Exam-style
A curve has the equation y = -2x^2 + 8x - 5. Find the coordinates of the maximum point and the range of y for the domain 0 ≤ x ≤ 5.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Sketching and range for a given domain
A sketch shows the turning point, the y-intercept (0, c) and any x-intercepts. For a restricted domain, the range runs from the lowest to the highest value of f on that domain. Check the turning point (if it is inside the domain) and both end points.
🧑🏫 Teacher's explanation
Think of a quadratic function as a roller coaster track. The shape is a parabola, which either looks like a valley (minimum) or a hill (maximum). When we talk about the 'range', we are simply asking: 'What are the highest and lowest heights the roller coaster reaches during this specific part of the ride?'
📖 Definitions
🔍 Worked example — Easy
Find the range of f(x) = x^2 - 4x + 5 for the domain 0 ≤ x ≤ 5.
🔍 Worked example — Standard
Find the range of g(x) = -2x^2 + 8x - 3 for the domain 1 ≤ x ≤ 4.
🔍 Worked example — Exam-style
A function is defined by h(x) = x^2 + 6x + 5 for -4 ≤ x ≤ 1. Determine the range.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The discriminant
For ax^2 + bx + c = 0:
🧑🏫 Teacher's explanation
Imagine you are throwing a ball into the air. The path it follows is a parabola. The 'roots' of the equation are the points where the ball hits the ground. Sometimes the ball hits the ground at two different spots (two real roots), sometimes it just barely grazes the ground at one single spot (two equal roots), and sometimes it never touches the ground at all (no real roots).
📖 Definitions
🔍 Worked example — Easy
Determine the nature of the roots for the equation x^2 - 5x + 6 = 0.
🔍 Worked example — Standard
Find the value of k for which the line y = kx - 5 is a tangent to the curve y = x^2 + 3x + 1.
🔍 Worked example — Exam-style
Show that the line y = 2x + c does not intersect the curve y = x^2 + 4x + 10 for any real value of c.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Solving quadratic equations
Factorise when you can; otherwise use the formula (given in the List of formulas) x = -b ± √(b^2 - 4ac)/2a, or complete the square. Paper 1 is non-calculator, so leave answers as exact surds where they do not simplify.
🧑🏫 Teacher's explanation
Solving a quadratic equation is like finding the exact moments when a ball, thrown into the air, hits the ground. If the ball's path is described by y = ax^2 + bx + c, the 'roots' are the values of x where the height y is zero. Depending on how high you throw it and the gravity acting on it, the ball might hit the ground twice, just graze the ground once, or never touch it at all.
📖 Definitions
🔍 Worked example — Easy
Solve x^2 - 5x + 6 = 0 by factorising.
🔍 Worked example — Standard
Solve 2x^2 + 4x - 3 = 0 using the quadratic formula.
🔍 Worked example — Exam-style
Determine the nature of the roots for 3x^2 - 2x + 5 = 0 and find the roots if they exist.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Quadratic inequalities
Find the critical values (the roots), sketch the parabola, and choose the region:
🧑🏫 Teacher's explanation
Solving a quadratic inequality is like finding which parts of a roller coaster track are above or below ground level. Instead of just finding the specific points where the track hits the ground (the roots), we want to find the entire stretches of track that are either entirely above the ground (> 0) or entirely below it (< 0).
📖 Definitions
🔍 Worked example — Easy
Solve the inequality x^2 - 5x + 4 > 0.
🔍 Worked example — Standard
Solve the inequality x^2 + 2x - 8 ≤ 0.
🔍 Worked example — Exam-style
Find the range of values of x for which 2x^2 - 8x + 6 < 0.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Find the values of m for which y = mx - 1 is a tangent to y = x^2 + 3.
🔍 Worked example — Easy
Find the values of m for which the line y = mx - 1 is a tangent to the curve y = x^2 + 3.
🔍 Worked example — Standard
Find the set of values of k for which the line y = 2x + k does not meet the curve y = x^2 - 4x + 7.
🔍 Worked example — Exam-style
Given f(x) = 5 + 6x - x^2, find the maximum value of f(x) and the range for the domain x ≥ 0.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) (x - 5)^2 - 4; minimum -4. (2) (x - 3)^2 - 7: minimum -7 at x = 3; f(0) = 2, f(7) = 9; range -7 ≤ f(x) ≤ 9. (3) 4k^2 - 12k = 0 ⇒ k = 3 (k = 0 is not a quadratic). (4) (3x - 1)(x + 2) ≥ 0: x ≤ -2 or x ≥ tfrac13. (5) x^2 - 3x + 2 = 0, discriminant 9 - 8 = 1 > 0, so two points: (1, 5) and (2, 6).