HSC Extension 1 and 2 Mathematics/The quadratic polynomial and the parabola

It is the purpose of this section to develop the algebraic properties of the quadratic function and to relate these to the parabolic curve.

The quadratic polynomial ax² + bx + c

The quadratic polynomial: ax² + bx + c is a quadratic polynomial of the second degree or a quadratic expression. To distinguish x from the coefficients a, b, and c, it may be called an indeterminate.

When the domain of x is specified, the quadratic polynomial becomes a function. In all quadratic polynomials to be studied, the coefficients will be rational (usually integers) and the domain of x will be the set of real numbers. The quadratic function will be expressed as

$y=ax^{2}+bx+c$

Graph of a quadratic function

Graphs of quadratic functions: very simple examples will have already been studied. In giving further practice in graphing quadratic functions the teacher should stress, in each particular case, points of general interest, e.g. (1) that for large values of x the term ax² effectively determines the value of the function; (2) the relation between the graph and the roots of the quadratic equation ax² + bx + c = 0. Examples should include cases where the graph has respectively two points, one point, and no points in common with the x-axis.

Roots of a quadratic equation

A value of x which makes y = 0 is a root of the quadratic equation ax2 + bx + c = 0. The term ‘zero of the polynomial’ might be introduced at the discretion of the teacher.

Quadratic inequalities

Quadratic inequalities: the graph of the quadratic function should be used to solve quadratic inequalities, e.g. find the values of m for which 12 + 4m – m² > 0.

General theory of quadratic equations, relation between roots and coefficients

Revision of simple quadratic equations which can be solved by factorisation.

Solution by ‘completing the square’ in particular cases.

It will be noted that, applied to an equation such as x² + 2x + 2 = 0 the method leads to (x + 1)² + 1 = 0, showing that no (real) value of x can be found which willmake x² + 2x + 2 equal to zero. The traditional formula is derived by applying the method of completing the square to the general quadratic.

The relation α + ß = –b/a, αß = c/a between the roots α, ß of a quadratic equation and its coefficients a, b, c, can be derived directly from the general solution. If an equation has roots α, ß, then it is of the form a(x–α) (x–ß) = 0 or a[x2 – (α + ß)x + αß] = 0.

Exercises involving finding equations whose roots bear stated relations to the roots of some other equation are not included in this syllabus.

The discriminant

The discriminant is to be defined and used to determine the condition for real, equal, or rational roots; pupils should be reminded of the meaning of the word ‘discriminate’ in ordinary language. By actually solving the general quadratic, an important existence theorem has been established: A quadratic equation may have two (real) roots, one root or no roots. It does not have more than two roots.

Classification of quadratic expressions; identity of two quadratic expressions

from $ax^{2}+bx+c=a\left[\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {4ac-b^{2}}{4a^{2}}}\right]$

(the general result would of course be preceded by particular examples), the conditions for positive definite, negative definite and indefinite quadratic expressions are derived. Only if b² > 4ac can the expression take both positive and negative values, and it has the same sign as a for all values of x except those lying between the roots of the equation ax² + bx + c = 0. Also, ax² + bx + c has its greatest or least value when x = – b/2a and this greatest or least value is (4ac – b²)/4a.

An alternative treatment is to consider the roots of the expression

$f(x)=ax^{2}+bx+c,a\neq 0$

1. Suppose the discriminant Δ = b² – 4ac < 0. Then f cannot be zero. Thus if Δ < 0 and a > 0, then f > 0 for all values of x, and is called positive definite. If Δ < 0 and a < 0, then f < 0 for all values of x, and is called negative definite. 2. If Δ > 0, then f = 0 for two distinct values of x, say x₁ and x₂. The greatest or least value of f occurs at x = 1/2 × (x₁ + x₂) and f takes both positive and negative values. 3. If Δ = 0, then f = 0 for one value of x, at x = – b/2a. Then f >= 0 if a > 0, and f <= 0 if a < 0, for all values of x. 4. the turning point of f is at df/dx = 0 (when the gradient of f is 0), i.e. at x = – b/2a.

Students should learn to find the turning point and zeros (if any) of f in order to sketch the graph of f.

Theorem: If a₁x² + b₁x + c₁ = a₂x² + b₂x + c₂ for more than two values of x, then

$a_{1}=a_{2},b_{1}=b_{2},c_{1}=c_{2}.$

The proof reduces to a discussion of the equation ax² + bx + c = 0 with a = a₁ – a₂, b = b₁ – b₂, and c = c₁ – c₂. Beginners find the proof elusive. Work on quadratic equations has shown that ax² + bx + c = 0 can exist for at most two values of x. There is one exception: if a = b = c = 0, the expression exists for all values of x. If it is given that ax² + bx + c = 0 for more than two values of x, we must conclude that a = b = c = 0. Otherwise the data presents us with a contradiction. Examples should include the expression of a quadratic polynomial ax² + bx + c in the form Ax(x – 1) + Bx + C, where C = c, A = a, B = a + b, the fitting of a quadratic to three given function values, and similar identities.

Equations reducible to quadratics

Examples of the following kinds should be discussed

1. $x^{4}-4x^{2}-12=0$
2. $(x+1)^{2}=4x^{2}$
3. $9^{x}-4(3)^{x}+3=0$
4. $(x+{\frac {1}{x}})^{2}-5(x+{\frac {1}{x}})+6=0$

The parabola defined as a locus

A parabola is defined as the locus of a point which moves so that its distance from a fixed point is equal to its distance from a fixed line.

Definitions of focus, directrix, vertex, axis and focal length should be given and illustrated by examples.

The equation x^2 = 4Ay

If the fixed point is (0, A) and the fixed line is y = –A, the equation of the locus is

$x^{2}=4Ay$ or $y=x^{2}/4A$

Use of change of origin when vertex is not at (0, 0)

By considering, for example, cases where: (a) the focus is (x₀, A) and the directrix is y = –A, (b) the focus is (0, y₀ + A) and the directrix is y = y₀ –A, (c) the focus is (x₀, y₀ + A) and the directrix is y = y₀ –A,

the interpretation of the equation

$(x-x_{0})^{2}=4A(y-y_{0})$

as representing a parabola with vertex (x₀, y₀), axis x = x₀, focus (x₀, y₀ + A) and directrix y = y₀ – A should be treated. Similarly, the equations (x – x₀)² = –4A(y – y₀), (y – y₀)² = 4A(x – x₀), (y – y₀)² = –4A(x – x₀) should be discussed.

Starting with the general quadratic function

$f(x)=ax^{2}+bx+c,a\neq 0$

and rewriting it as

$\left(x+{\frac {b}{2a}}\right)={\frac {1}{a}}\left(y+{\frac {b^{2}-4ac}{4a}}\right)$

Practice should also be given in finding the equation of a parabola given for example, its vertex, axis and focal length, or in finding the equation of the family of parabolas having, for example, the line x = x0 as axis and a given vertex or focal length or passing through a given point.