Quadratic Functions and Equations for the TMUA

Q: What does a negative discriminant mean for the graph of a quadratic?

A negative discriminant ($b^2 - 4ac 0$) or strictly below the $x$ axis (if $a < 0$).

Q: How do I find the range of a quadratic function?

The range is determined by the $y$ coordinate of the vertex. For $a > 0$, the range is $y \geq q$, and for $a < 0$, the range is $y \leq q$, where $q$ is the $y$ value found after completing the square.

Q: When should I factorise instead of using the formula?

Factorisation is generally faster if the roots are integers or simple fractions. If you cannot spot factors within a few seconds, it is usually more efficient to proceed with the quadratic formula or completing the square.

Updated August 2025

This lesson covers the properties of quadratic functions, their graphs, and the essential algebraic techniques required for the TMUA. You will learn to manipulate quadratics via factorisation, completing the square, and the quadratic formula, while understanding how the discriminant determines the nature of the roots.

Core concept

A quadratic function is a polynomial of the form $f(x) = ax^2 + bx + c$ where $a \neq 0$ , characterised by a parabolic graph and roots determined by the discriminant $b^2 - 4ac$ .

Defining Quadratic Functions

Quadratics are functions generally written in the form $ax^2 + bx + c$ where $a \neq 0$ . The term quadratic is often used loosely to refer to the function itself, its graph, or the algebraic expression. In university admissions tests like the TMUA, quadratics serve as the perfect toy function: they are simple enough to manipulate while possessing rich properties that illustrate deeper mathematical concepts such as symmetry and rates of change.

To master this topic, you are expected to be able to:

Factorise quadratics when appropriate.
Solve equations using the quadratic formula.
Complete the square and understand its relationship to the quadratic formula.
Understand the link between the roots and the graph.
Relate the completed square form to transformations on the $xy$ plane.
Sketch curves, marking $x$ intercepts, $y$ intercepts, and vertex coordinates.
Use the discriminant to determine the number of real roots.
Find maximum or minimum coordinates via completing the square or differentiation.

Completing the Square and Graph Shifting

Consider the simplified quadratic $y = x^2 + bx + c$ . By completing the square, we can rewrite this as:

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

This form reveals that the graph is simply the basic parabola $y = x^2$ shifted, or translated, on the $xy$ plane. Specifically, starting with $y = x^2$ , we shift it to the left by $\frac{b}{2}$ and up by $-\frac{b^2}{4} + c$ . This transformation does not stretch or squash the shape, it merely moves it.

We can find the coordinates of the minimum point of $y = x^2 + bx + c$ using three distinct methods:

Translation: The minimum of $y = x^2$ is $(0, 0)$ . Shifting it as described gives the new minimum at $\left(-\frac{b}{2}, -\frac{b^2}{4} + c\right)$ .
Differentiation: Setting $\frac{\mathrm{d}y}{\mathrm{d}x} = 2x + b = 0$ yields $x = -\frac{b}{2}$ . Substituting this back into the original equation gives the $y$ coordinate.
Algebraic Inspection: Since $\left(x + \frac{b}{2}\right)^2$ is always greater than or equal to zero, the least possible $y$ value occurs when this squared term is exactly zero, which happens when $x = -\frac{b}{2}$ .

The Discriminant and Roots

The roots of a quadratic are the $x$ values where $y = 0$ , representing the points where the graph crosses the $x$ axis. For a graph with a positive $x^2$ coefficient, it only crosses the $x$ axis if its minimum point is below the axis. Algebraically, this requirement is $- \frac{b^2}{4} + c < 0$ , which rearranges to $b^2 - 4c > 0$ .

For the general quadratic $ax^2 + bx + c$ , the nature of the roots is determined by the discriminant, $b^2 - 4ac$ :

If $b^2 - 4ac > 0$ , the quadratic has two real distinct roots and cuts the $x$ axis at two points.
If $b^2 - 4ac = 0$ , the quadratic has one repeated root and touches the $x$ axis at a single point.
If $b^2 - 4ac < 0$ , the quadratic has no real roots and never touches or crosses the $x$ axis.

The General Quadratic Formula

Completing the square on $ax^2 + bx + c = 0$ allows us to derive the quadratic formula. We start by dividing by $a$ :

$a \left( (x + \frac{b}{2a})^2 - \frac{b^2}{4a^2} + \frac{c}{a} \right) = 0$

Rearranging to isolate $x$ :

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

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

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

This formula provides the horizontal positions of the roots. The line of symmetry for the graph is at $x = -\frac{b}{2a}$ , and the distance from this line to each root is $\frac{\sqrt{b^2 - 4ac}}{2a}$ . Consequently, the total distance between the roots is $\frac{\sqrt{b^2 - 4ac}}{a}$ .

Alternative Forms and Exercises

Sometimes, an alternative version of the quadratic formula is useful. If we divide the original equation by $x^2$ , we create a quadratic in terms of $\frac{1}{x}$ . Solving this leads to:

$x = \frac{2c}{-b \pm \sqrt{b^2 - 4ac}}$

While this is equivalent to the standard formula, it requires $c \neq 0$ . Thinking about such variations helps develop the flexibility needed for the TMUA.

Key takeaways

The discriminant $b^2 - 4ac$ determines if there are two, one, or zero real roots based on its sign.
The completed square form $a(x+p)^2 + q$ reveals the vertex of the parabola at $(-p, q)$ .
The quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ is derived directly from the process of completing the square.
The line of symmetry for any quadratic $ax^2 + bx + c$ is always the vertical line $x = -\frac{b}{2a}$ .

Tips

Always check for a common factor across $a$ , $b$ , and $c$ before solving, as this simplifies the arithmetic significantly. When using the discriminant in inequalities, remember that a quadratic 'touching' the axis counts as having a repeated root ( $b^2 - 4ac = 0$ ).

Cautions

Be extremely careful with signs when calculating the discriminant, especially when $c$ is negative. A common error is writing $-4ac$ as a subtraction when it should become an addition.

Insight

The relationship between the coefficients and the roots (Vieta's Formulas) is a powerful tool: the sum of the roots is $-b/a$ and the product of the roots is $c/a$ . This often allows you to solve TMUA problems without ever finding the roots explicitly.

Worked Examples

Example 1

How many real roots does the equation

x^4 - 4x^3 + 4x^2 - 1 = 0

have?

A:0

B:1

C:2

D:3

E:4

Practice Questions

Practice Question 1

f(x)

is a quadratic function in

x

. The graph of

y = f(x)

passes through the point

(1, -1)

and has a turning point at

(-1, 3)

. Find an expression for

f(x)

-x^2 - 2x + 2

-x^2 + 2x + 3

x^2 - 2x

x^2 + 2x - 4

2x^2 + 4x + 1

-2x^2 - 4x + 5

Frequently asked questions

What does a negative discriminant mean for the graph of a quadratic?

A negative discriminant ( $b^2 - 4ac < 0$ ) means the quadratic has no real roots. Geometrically, the entire parabola sits either strictly above the $x$ axis (if $a > 0$ ) or strictly below the $x$ axis (if $a < 0$ ).

How do I find the range of a quadratic function?

The range is determined by the $y$ coordinate of the vertex. For $a > 0$ , the range is $y \geq q$ , and for $a < 0$ , the range is $y \leq q$ , where $q$ is the $y$ value found after completing the square.

When should I factorise instead of using the formula?

Factorisation is generally faster if the roots are integers or simple fractions. If you cannot spot factors within a few seconds, it is usually more efficient to proceed with the quadratic formula or completing the square.

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