Quadratic Functions, SPM Additional Mathematics Form 4

A quadratic function is any rule of the form $f(x)=ax^{2}+bx+c$ where $a$, $b$ and $c$ are constants and $a\neq 0$. The condition $a\neq 0$ matters: if $a$ were zero the $x^{2}$ term would disappear and the rule would only be linear.

Because of that squared term, the graph is never a straight line, it is a smooth U-shaped curve called a parabola. Quadratic Functions is Chapter 2 of Form 4 Add Math, and it is the chapter where you move from the general idea of a function, met in Chapter 1, to a specific family of functions you can solve, factorise, sketch and optimise.

The chapter carries three content standards taken directly from the KSSM DSKP: Quadratic Equations and Inequalities, Types of Roots of Quadratic Equations, and Quadratic Functions themselves. Together they cover four connected skills, solving a quadratic equation, judging how many real roots it has from the discriminant, building an equation back up from its roots, and describing or sketching the parabola that the function draws.

None of these stands alone; each one leans on the others, which is why we teach the chapter as a single connected story rather than a list of tricks.

Our teachers treat Quadratic Functions as a workhorse chapter. The algebra you rehearse here, expanding, factorising, completing the square, substituting into the formula, turns up again and again for the rest of the course.

Because SPM uses analytic scoring, you are paid for every correct line of working, not only for the final answer, so a tidy, well-set-out solution to a quadratic is worth practising until it is automatic. A student who can solve a quadratic three different ways, and choose the quickest for the question in front of them, has a genuine advantage in the exam room.

It helps to keep the parabola picture in mind throughout. When $a>0$ the curve opens upwards and has a lowest point; when $a<0$ it opens downwards and has a highest point.

That single lowest or highest point, the vertex, is the key to maximum and minimum problems, and it is why the vertex form $a(x-h)^{2}+k$ is so useful. Almost every question in the chapter is, underneath, a question about where the parabola sits, which way it opens, and where it crosses the axes.

The pay-off arrives quickly and keeps arriving. Quadratic thinking reappears in Systems of Equations when a linear and a non-linear equation meet, in Coordinate Geometry when a line meets a curve, and in Differentiation when you locate turning points.

Maximum-and-minimum word problems, the profit that is largest, the area that is greatest, the height that is highest, are quadratic problems in disguise. Because so much of the later syllabus quietly assumes you are fluent here, we never rush past this chapter; the time you invest now is repaid many times over.

The Quadratic Functions chapter is organised into three content standards. The codes and titles below come directly from the DSKP, learn them so you can recognise at a glance exactly what a question is testing.

Notice how the three standards build on one another. Standard 2.1 gives you the tools to solve and to build quadratic equations; 2.2 adds the discriminant, which lets you judge the roots without solving; and 2.3 lifts the whole picture onto a graph, where roots become x-intercepts and the vertex becomes a maximum or minimum.

If your solving in 2.1 is shaky, everything after it feels harder than it should, so make 2.1 solid first.

What a quadratic function is. The rule $f(x)=ax^{2}+bx+c$ with $a\neq 0$ always draws a parabola.

The sign of $a$ decides the direction: $a>0$ opens upwards (a minimum point), $a<0$ opens downwards (a maximum point). The size of $a$ decides how narrow or wide the curve is, and the constant $c$ is the y-intercept, the value of $f(0)$.

Solving by completing the square. Any quadratic can be rewritten as a perfect square plus a constant.

Factor out $a$ from the first two terms, add and subtract the square of half the resulting coefficient of $x$, and tidy up. This method not only solves the equation, it also hands you the vertex directly, which is why it is worth mastering even when the formula would be faster.

Solving by formula. When factorising is awkward, the quadratic formula solves $ax^{2}+bx+c=0$ every time.

This is one of the formulae supplied on the SPM formula list, so you do not have to memorise it, but you must be able to substitute into it cleanly, watching every sign.

$$x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$

The discriminant. The expression under the root sign, $b^{2}-4ac$, is called the discriminant, and on its own it tells you how many real roots the equation has before you solve anything.

This decision rule is not on the supplied formula list, so learn it: $b^{2}-4ac>0$ gives two distinct real roots, $b^{2}-4ac=0$ gives two equal (repeated) real roots, and $b^{2}-4ac<0$ gives no real roots.

$$b^{2}-4ac \;\begin{cases} >0 & \text{two distinct real roots}\\ =0 & \text{two equal real roots}\\ <0 & \text{no real roots}\end{cases}$$

Forming an equation from its roots. If $\alpha$ and $\beta$ are the roots, then the sum of roots is $\alpha+\beta=-\dfrac{b}{a}$ and the product is $\alpha\beta=\dfrac{c}{a}$.

From these you can rebuild the equation. These relationships are not on the supplied formula list, so they must be remembered.

$$x^{2}-(\alpha+\beta)x+\alpha\beta=0$$

Quadratic inequalities. To solve something like $x^{2}-x-6>0$, first find the roots of the matching equation, then decide which region satisfies the inequality.

A quick sketch of the parabola makes this almost visual: for a "$>0$" inequality with $a>0$ you want the parts of the curve above the x-axis (outside the roots); for "$<0$" you want the part below (between the roots). A number line or a sign table gives the same answer if you prefer not to sketch.

The vertex form. Completing the square rewrites $f(x)=ax^{2}+bx+c$ as $f(x)=a(x-h)^{2}+k$.

In this form the vertex is read off immediately as $(h,k)$, and the axis of symmetry is the vertical line $x=h$. This form is the fastest route to a sketch and to any maximum or minimum value.

$$f(x)=a(x-h)^{2}+k \quad\Rightarrow\quad \text{vertex }(h,\,k),\ \text{axis }x=h$$

Effects of $a$, $h$ and $k$. In vertex form, changing $k$ slides the parabola up or down, changing $h$ slides it left or right, and changing $a$ stretches it or flips it over.

Understanding these three movements lets you picture any quadratic without plotting a single point.

Graph position and the roots. Because the roots are where $f(x)=0$, the discriminant also tells you how the parabola meets the x-axis: two distinct roots means it cuts the axis at two points, one repeated root means it just touches the axis at the vertex, and no real roots means it never meets the axis at all.

This is the bridge that ties standard 2.2 to standard 2.3.

Maximum and minimum values. The vertex is the maximum point when $a<0$ and the minimum point when $a>0$.

Real-life problems, largest area, greatest profit, lowest cost, usually reduce to finding this vertex, so completing the square is often the whole solution.

Quadratic Functions can appear in either written paper. Paper 1 (3472/1) lasts 2 hours and carries 80 marks: Section A has 12 questions worth 64 marks that you answer all of, and Section B has 3 questions worth 16 marks from which you answer 2.

Paper 2 (3472/2) lasts 2 hours 30 minutes and carries 100 marks across Section A (7 questions, 50 marks, answer all), Section B (4 questions, 30 marks, answer 3) and Section C (4 questions, 20 marks, answer 2).

Across both papers the items are limited-response subjective and structured questions, they are marked using analytic scoring, and you sit them with a non-programmable scientific calculator. We do not predict how many marks any single chapter will carry, because that varies from year to year, but quadratic skills support many questions beyond this chapter, so being fluent here helps well outside Chapter 2.

In practice a quadratic item is usually structured: a short stem gives you an equation or a function, then successive parts ask you to solve it, discuss its roots through the discriminant, complete the square, or sketch and interpret the graph. Answer the parts in order and carry your earlier results forward carefully, the vertex you find in one part is often exactly what the next part needs.

A clean, labelled sketch, even when only a rough one is asked for, protects the marks that follow it.

Most marks lost in Quadratic Functions come from a small set of avoidable habits. Read these before every practice set until they become second nature.

• Sign slips in the discriminant. $b^{2}$ is never negative, even when $b$ is negative, and the term is $-4ac$ with its own sign. Compute $b^{2}-4ac$ carefully, a single sign error here changes the type of roots and loses the whole part.
• Confusing equal roots with no roots. $b^{2}-4ac=0$ means two equal (repeated) real roots and a curve that touches the x-axis; $b^{2}-4ac<0$ means no real roots and a curve that misses the axis entirely. Match the condition to the wording exactly.
• Dropping the minus in the sum of roots. The sum of roots is $\alpha+\beta=-\frac{b}{a}$, not $\frac{b}{a}$. Forgetting that minus sign is one of the most common errors when forming an equation from its roots.
• Not factoring out $a$ before completing the square. When $a\neq 1$ you must factor $a$ from the first two terms first, or the perfect square you build will be wrong. Keep $a$ outside the bracket until the very end.
• Reading the wrong region of a quadratic inequality. Whether the solution is between the roots or outside them depends on the inequality sign and on the sign of $a$. Sketch the parabola or use a sign table rather than guessing.
• Getting the sign of $h$ wrong in vertex form. In $a(x-h)^{2}+k$ the vertex is $(h,k)$, so $a(x+3)^{2}+1$ has vertex $(-3,1)$, not $(3,1)$. Read the bracket as $(x-h)$ and take $h$ with the opposite sign to what you see.

None of these is about ability, each is simply a habit, and habits are fixable. Tick them off one at a time in your practice, and your accuracy in this chapter climbs quickly.

The students who score well here are rarely the fastest; they are the ones who are consistently tidy, so treat every slip as feedback rather than failure.

Quadratic Functions rewards a steady, ordered approach. Work through the steps below in order, then use the resources that follow to revise and to test yourself under time.

Aim for short, frequent sessions rather than one long cram. A couple of solved equations, one completed square and one sketch each day keep the skills fresh and steadily wear away the small slips that cost marks.

When a method starts to feel automatic, move on to the next; when it does not, slow down and repeat it until it does. Spread across a couple of weeks, this quiet routine turns Chapter 2 into some of your most dependable marks.

1. 1

   Solve three ways

   Take one quadratic and solve it by factorising, by completing the square, and by formula, so you can choose the fastest route for any question.

2. 2

   Make the discriminant automatic

   Drill $b^{2}-4ac$ and its three cases until you can name the type of roots at a glance, and practise the 'range of values' questions that use it.

3. 3

   Link roots and coefficients

   Rehearse $\alpha+\beta=-\frac{b}{a}$ and $\alpha\beta=\frac{c}{a}$, then form new equations from given roots and check by expanding.

4. 4

   Master the vertex form

   Complete the square to reach $a(x-h)^{2}+k$, read off the vertex and axis of symmetry, and use it for maximum and minimum problems.

5. 5

   Sketch, then test under time

   Practise quick, labelled sketches, direction, vertex, intercepts, then attempt mixed practice and worked examples with a clock running.

• Quadratic Functions, Revision Notes
• Quadratic Functions, Common Mistakes
• Quadratic Functions, Practice Questions
• Quadratic Functions, Paper 2 Answering Guide
• Quadratic Functions, Key Terms
• Quadratic Functions, Worked Examples (Easy)
• Quadratic Functions, Worked Examples (Medium)
• Quadratic Functions, Worked Examples (KBAT)