CAIE IGCSE Additional Math 2 quadratic functions

Use this Quadratic Functions hub to move from completed-square form and vertex values into discriminants, equations and quadratic inequalities.

Covered syllabus points

2.1. Find maximum and minimum values 2.2. Use vertex values to sketch or find range 2.3. Use discriminant conditions 2.4. Solve quadratic equations 2.5. Solve quadratic inequalities

[Maximum number: 3]

(a) Find the coordinates of the stationary point on the curve y=(x+3)(x-4).

Find the coordinates of the stationary point on the curve y=(x+3)(x-4).

$\begin{aligned} y=x^{2}-x-12 \frac{\mathrm{~d} y}{\mathrm{~d} x}=2 x-1 \text { or }(x+3)+(x-4) \text { or } y=\left(x-\frac{1}{2}\right)^{2}-\frac{49}{4} \text { or using symmetry } x=\frac{4-3}{2} \end{aligned}$

M1

For expanding the brackets and
differentiate with at least one correct term
or for using the product rule
or for completing the square
or for using symmetry

$x=\frac{1}{2}$

A1

$y=-\frac{49}{4}$ oe

A1

[Maximum number: 3]

The function f is defined by $\mathrm{f}(x)=1-4 x-x^{2}$ for all real values of x.

(a)

Write f(x) in the form $a-(x+b)^{2}$ , where a and b are constants.

[ 2 ]

$5-(x+2)^{2}$

B2

B1 for $-(x+2)^{2}$ or $(x+2)^{2}$

or a=5 and b=2

(b)

Find the range of f.

[ 1 ]

$\mathrm{f} \leqslant 5$ or $\mathrm{f}(x) \leqslant 5$

B1

FT their 5

[Maximum number: 1]

The polynomial p is given by $\mathrm{p}(x)=a^{2} x^{3}+2 a x^{2}+a x+2$ , where a is a positive integer. It is given that 2 x+1 is a factor of p(x).

Hence show that the equation p(x)=0 has only one real root.

Discriminant of $4 x^{2}+1$ is 0-16<0 oe and no real roots oe or $4 x^{2}+1=0 \rightarrow 4 x^{2}=-1$ oe and therefore no solution oe

[statement that only solution is $x=-\frac{1}{2}$ ]

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Covered syllabus points

2. Quadratic functions question 1

2. Quadratic functions question 2

Question (a)

Question (b)

2. Quadratic functions question 3