IP Maths Notes (Lower Sec, Year 1-2): 05) Quadratic Expressions & Graph Sketching

Quadratics surface in projectile motion, optimisation, and curve sketching. Build muscle memory for algebraic techniques and graphical meaning.

Learning targets

• Factorise simple quadratics and solve \( ax^2 + bx + c = 0 \).
• Complete the square to identify vertex form.
• Determine symmetry, intercepts, and turning points.
• Sketch parabolas with annotated intercepts and axis of symmetry.

1. Factorisation recap

Use the product-sum method for monic quadratics: find numbers \( p, q \) such that \( p + q = b \) and \( pq = c \).

Example: Factorise \( x^2 - 5x + 6 \) → \( (x - 2)(x - 3) \).

For non-monic forms, apply grouping or the quadratic formula.

2. Completing the square

Convert \( ax^2 + bx + c \) into \( a[(x - h)^2 - k] \) form.

Example: Rewrite \( y = x^2 - 6x + 5 \).

\[ y = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4. \]

Vertex at \( (3, -4) \). Axis of symmetry: \( x = 3 \).

For coefficient \( a \neq 1 \), factor out before completing: \( ax^2 + bx = a[x^2 + \frac{b}{a}x] \).

3. Quadratic formula

Use when factorisation is not obvious:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]

Discriminant \( D = b^2 - 4ac \) indicates nature of roots:

• \( D > 0 \) → two distinct real roots.
• \( D = 0 \) → one repeated real root.
• \( D < 0 \) → no real roots.

4. Graph sketching workflow

1. Find intercepts: set \( y = 0 \) for x-intercepts, \( x = 0 \) for y-intercept.
2. Complete the square to find vertex.
3. Plot axis of symmetry and key points.
4. Sketch the curve, indicating direction (opens up if \( a > 0 \)).

Worked example — Sketching \( y = -2x^2 + 8x - 5 \)

• Axis: complete the square.

\[ y = -2[x^2 - 4x] - 5 = -2[(x - 2)^2 - 4] - 5 = -2(x - 2)^2 + 8 - 5 = -2(x - 2)^2 + 3. \]

Vertex: \( (2, 3) \); axis: \( x = 2 \).

• Y-intercept: substitute \( x = 0 \) → \( y = -5 \).
• X-intercepts: solve \( -2x^2 + 8x - 5 = 0 \).

\[ x = \frac{-8 \pm \sqrt{64 - 4(-2)(-5)}}{-4} = \frac{-8 \pm \sqrt{64 - 40}}{-4} = \frac{-8 \pm \sqrt{24}}{-4}. \]

Since \( \sqrt{24} = 2\sqrt{6} \):

\[ x = \frac{-8 \pm 2\sqrt{6}}{-4} = 2 \mp \frac{\sqrt{6}}{2}. \]

Plot these approximate values for accuracy.

5. Optimisation context

Quadratics model maximum/minimum scenarios. Example: The area of a rectangle with fixed perimeter \( P \) is maximised when it is a square. Algebraically, express area \( A = x(P/2 - x) \), complete the square, and show vertex at \( x = P/4 \).

Try it yourself

1. Factorise \( 3x^2 - 7x - 6 \) and solve \( 3x^2 - 7x - 6 = 0 \).
2. Complete the square for \( y = 2x^2 + 8x + 3 \) and state the minimum value.
3. Sketch \( y = x^2 - 4x - 5 \), labelling intercepts and vertex.
4. A product has revenue \( R(x) = -5x^2 + 150x \). Find the quantity \( x \) that maximises revenue and the maximum value.

Move on to geometry at https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-06-Geometry-Congruency-and-Similarity.