Introduction to Quadratic Functions

Welcome to the world of Quadratic Functions! While the name might sound a bit intimidating, you’ve actually been seeing these shapes your whole life. Ever watched a basketball arc through the air toward the hoop? Or looked at the cables of a suspension bridge? Those beautiful curves are parabolas, and they are described by quadratic functions.

In this chapter, we are going to learn how to "detect" the roots of these functions, how to find their lowest or highest points, and how to solve equations that are secretly quadratics in disguise. Don't worry if algebra feels like a puzzle—we’re going to solve it one piece at a time!


1. What is a Quadratic Function?

A quadratic function is any expression that can be written in the standard form:
\(y = ax^2 + bx + c\)

Key things to remember about this form:
1. \(x^2\) is the star: The highest power of \(x\) must be 2.
2. \(a, b,\) and \(c\) are just numbers (called constants).
3. \(a\) cannot be zero: If \(a\) were 0, the \(x^2\) term would disappear, and we’d just have a straight line!

The Shape of the Graph

The graph of a quadratic is a curve called a parabola. You can tell a lot about the graph just by looking at the first number, \(a\):

  • If \(a\) is positive (\(a > 0\)): The curve is a "smiley face" (u-shape). It has a minimum point at the bottom.
  • If \(a\) is negative (\(a < 0\)): The curve is a "frowning face" (n-shape). It has a maximum point at the top.

Quick Review: Think "Positive people smile, Negative people frown." This will help you remember which way the curve opens!


2. The Discriminant: The "Root Detective"

Before we even solve a quadratic equation, we can find out what *kind* of answers (roots) we will get. We do this using a special part of the quadratic formula called the discriminant.

The discriminant is usually written as \(D\) or the Greek letter \(\Delta\) (Delta), and the formula is:
\(D = b^2 - 4ac\)

There are three possible outcomes for the discriminant, and each tells us something different about the graph and its roots (where the graph crosses the x-axis):

  1. If \(b^2 - 4ac > 0\): There are two real distinct roots. The graph crosses the x-axis at two different points.
  2. If \(b^2 - 4ac = 0\): There is one repeated real root. The graph just touches the x-axis at one point and bounces back (the x-axis is a tangent).
  3. If \(b^2 - 4ac < 0\): There are no real roots. The graph stays entirely above or entirely below the x-axis and never touches it.

Did you know? In the third case (\(D < 0\)), the roots still exist in a world called "Complex Numbers," but for your AS Level exams, we simply say they are not real.

Key Takeaway: Always check the discriminant first if a question asks about the "nature of the roots" or if roots exist at all.


3. Completing the Square

Sometimes the standard form \(ax^2 + bx + c\) isn't very helpful for sketching. We can rewrite it into completed square form:
\(y = a(x + p)^2 + q\)

This form is like a "GPS" for your parabola. It tells you exactly where the turning point (the vertex) is.

How to find the Turning Point

If your equation is in the form \(y = a(x + p)^2 + q\):

  • The Turning Point is at \( (-p, q) \). (Notice the sign of p flips!)
  • The Line of Symmetry is the vertical line \( x = -p \).

Example: If \(y = 2(x + 3)^2 + 4\):
The turning point is \( (-3, 4) \).
The line of symmetry is \( x = -3 \).
Because the \(2\) is positive, this is a minimum point (the bottom of a valley).

How to Complete the Square (Step-by-Step)

Let's look at \(x^2 + 6x + 10\):

1. Halve the middle number: Half of 6 is 3. Write it as \((x + 3)^2\).
2. Subtract the square: Square that 3 (which is 9) and subtract it: \((x + 3)^2 - 9\).
3. Add the end constant: Bring down the +10: \((x + 3)^2 - 9 + 10\).
4. Simplify: \(y = (x + 3)^2 + 1\).

Common Mistake: Forgetting to subtract the square of the half-number. Always subtract it, even if the middle number is negative!


4. Solving "Hidden" Quadratics

Sometimes, an equation doesn't look like a quadratic at first glance, but it follows the same pattern. These are called quadratics in a function of the unknown.

Example 1: \(x^4 - 5x^2 + 6 = 0\)
Notice that \(x^4\) is just \((x^2)^2\). If we replace \(x^2\) with a new letter, say \(u\), the equation becomes:
\(u^2 - 5u + 6 = 0\)
This is a standard quadratic! We solve for \(u\), then remember to swap back to find \(x\).

Example 2: \(x^{\frac{2}{3}} - 5x^{\frac{1}{3}} + 4 = 0\)
Here, we can let \(u = x^{\frac{1}{3}}\). The equation becomes \(u^2 - 5u + 4 = 0\).

Step-by-Step for Hidden Quadratics:

1. Identify the "Middle Function": Look at the middle term (the one without the square). Let \(u\) equal that function.
2. Rewrite: Replace the terms to get a quadratic in \(u\).
3. Solve: Find the values of \(u\) (by factorising or using the formula).
4. Substitute back: Set your original function equal to those \(u\) values and solve for \(x\).

Encouraging phrase: Don't worry if this seems tricky at first! It’s just like wearing a mask—once you take the mask off (the substitution), it’s the same old quadratic you already know.


5. Summary and Quick Review

The Essentials Checklist:

  • Standard Form: \(y = ax^2 + bx + c\)
  • The Discriminant (\(b^2 - 4ac\)): Tells you if there are 2, 1, or 0 real roots.
  • Turning Point: Found easily using Completed Square form \(a(x+p)^2 + q\).
  • Symmetry: Every parabola is perfectly symmetrical. The mirror line goes right through the turning point.
  • Hidden Quadratics: Use substitution (let \(u = \text{something}\)) to make the equation manageable.

Key Takeaway: Quadratics are all about balance and patterns. Whether you are looking at the discriminant to predict the roots or completing the square to find the turning point, you are simply looking for the underlying structure of the curve.