Functions

Introduction to Functions: Your Mathematical Vending Machine

Welcome to the world of Functions! If you’ve ever used a vending machine, you already understand how a function works. You put in a specific code (the input), and the machine follows a rule to give you a specific snack (the output). In Mathematics, a function is just a rule that connects an input value (usually \(x\)) to a single output value (usually \(y\) or \(f(x)\)).

In this chapter, we will explore how to describe these rules, how to draw them as graphs, and how to "tweak" them to change their shapes. Don't worry if it feels abstract at first—we'll break it down step-by-step!

1. Quadratic Functions: The Power of the Square

A Quadratic Function is any function where the highest power of \(x\) is 2. They look like this: \(f(x) = ax^2 + bx + c\). When you graph them, they always form a "U" or "n" shape called a parabola.

The Discriminant: The "Root" Finder

Before you even draw a quadratic, you can predict how many times it will cross the \(x\)-axis (its roots) using the Discriminant. This is the part of the quadratic formula found under the square root sign: \(b^2 - 4ac\).

• If \(b^2 - 4ac > 0\): There are two real, distinct roots (the graph crosses the \(x\)-axis twice).
• If \(b^2 - 4ac = 0\): There is one repeated root (the graph just touches the \(x\)-axis at one point).
• If \(b^2 - 4ac < 0\): There are no real roots (the graph floats above or sinks below the \(x\)-axis).

Quick Review Box:
Positive Discriminant = 2 Crosses
Zero Discriminant = 1 Touch
Negative Discriminant = 0 Crosses

Completing the Square

Rewriting a quadratic in the form \(y = a(x + p)^2 + q\) is incredibly useful for sketching. It tells you exactly where the "nose" of the graph is—this is called the turning point (or vertex).

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

Example: For \(y = (x - 3)^2 + 5\), the turning point is \((3, 5)\). The graph is a "U" shape with its lowest point at \(x = 3, y = 5\).

Common Mistake to Avoid: When finding the turning point from \( (x+p)^2 \), students often forget to change the sign of \(p\). If it's \((x-4)\), the coordinate is \(+4\)!

Key Takeaway: The Discriminant tells you "how many" roots; Completing the Square tells you "where" the graph turns.

2. Disguised Quadratics: The Imposters

Sometimes an equation doesn't look like a quadratic at first glance, but it's "hiding" in another function. These are called equations in a function of the unknown.

Example: \(x^4 - 5x^2 + 6 = 0\)
This looks like a power-of-4 equation, but if we pretend \(x^2\) is just a single letter, say \(u\), it becomes:
\(u^2 - 5u + 6 = 0\)
Now it’s a simple quadratic! Solve for \(u\), then remember to substitute \(x^2\) back in to find the final values for \(x\).

Step-by-Step Process:
1. Identify the "middle" term variable (e.g., \(x^2\) or \(\sqrt{x}\)).
2. Substitute a new letter like \(u\) for that variable.
3. Solve the quadratic for \(u\).
4. Set your variable back equal to those answers and solve for \(x\).

3. Curve Sketching: Drawing the Story

In your exam, there is a big difference between plotting and sketching. Plotting means using a table of values and being precise. Sketching means showing the general shape and labeling key points where the graph crosses the axes.

Polynomials (Cubic and Quartic)

For polynomials like cubic (\(x^3\)) or quartic (\(x^4\)) functions, follow these cues:

• Intercepts: Set \(x=0\) to find where it hits the \(y\)-axis. Set \(y=0\) (factorise) to find where it hits the \(x\)-axis.
• Repeated Roots: If a factor is squared, like \((x-2)^2\), the graph touches the \(x\)-axis at \(2\) and bounces back.
• End Behavior: A positive \(x^3\) graph starts low and ends high. A negative \(x^3\) starts high and ends low.

Reciprocal Graphs

You need to know two specific shapes:

• \(y = \frac{a}{x}\): This graph has two curves in opposite quadrants. It never touches the axes. These "forbidden" lines are called Asymptotes.
• \(y = \frac{a}{x^2}\): Since \(x^2\) is always positive, both "arms" of the graph stay above the \(x\)-axis, looking a bit like a volcano.

Did you know? The word Asymptote comes from a Greek word meaning "not falling together." The graph gets closer and closer to the line but never actually meets it—mathematical social distancing!

4. Solving Equations Graphically

If you have two equations, say \(y = f(x)\) and \(y = g(x)\), the points of intersection are where \(f(x) = g(x)\).

• To solve \(x^2 = \frac{1}{x}\) graphically, you would sketch \(y = x^2\) and \(y = \frac{1}{x}\) on the same axes.
• The \(x\)-coordinate where the two lines cross is your solution.

Key Takeaway: The solution to an equation is simply the "meeting point" of two graphs.

5. Graph Transformations: Moving the Curve

Think of transformations as using a remote control to move a graph around the screen. You only need to know how to do one transformation at a time for AS Level.

The "Outside" Changes (Vertical)

If the change is outside the brackets, it affects the y-coordinates and follows common sense.

• \(y = f(x) + a\): Translation (sliding) Up by \(a\).
• \(y = a f(x)\): Vertical Stretch by scale factor \(a\). (Multiply \(y\) by \(a\)).

The "Inside" Changes (Horizontal)

If the change is inside the brackets with the \(x\), it affects the x-coordinates and does the opposite of what you'd expect!

• \(y = f(x + a)\): Translation Left by \(a\). (Wait, plus means left? Yes! Inside is "weird").
• \(y = f(ax)\): Horizontal Stretch by scale factor \(\frac{1}{a}\). (Divide \(x\) by \(a\)).

Memory Aid: "In-X-Out-Y"
Inside the brackets affects X (and is Inverse/opposite).
Outside the brackets affects Y (and is exactly what it looks like).

Translating with Vectors:
A translation can be written as a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\).
So, \(y = f(x-3) + 2\) would be a translation by the vector \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\).

Summary Checklist

Before your exam, make sure you can:
1. Use the discriminant to find how many roots a quadratic has.
2. Complete the square to find the turning point of a parabola.
3. Spot a "disguised" quadratic and solve it using substitution.
4. Sketch polynomials and reciprocal graphs, labeling the asymptotes.
5. Apply single transformations (stretch or slide) to any given function.

Don't worry if transformations feel tricky at first. Just remember: if it's inside the brackets, do the opposite to the \(x\); if it's outside, do exactly what it says to the \(y\)!