Graphs and transformations

Welcome to the World of Graphs!

In this chapter, we are going to learn how to visualize mathematics. Graphs are like "maps" for functions; they show us exactly how numbers behave. Whether you are aiming for an A or just trying to get a solid grasp of the basics, this guide will break down the curves, the lines, and the "magic moves" (transformations) that bring them to life. Don't worry if this seems tricky at first—once you see the patterns, it becomes much easier!

1. Famous Families of Graphs

Before we can move graphs around, we need to recognize the "standard" shapes. In H2 Math, you'll encounter a few specific families of curves.

A. Parabolas (The "U" Shapes)

You already know \(y = ax^2\). In this syllabus, we also look at sideways parabolas:
Standard Form: \(y^2 = ax\) or \(x^2 = by\)
Tip: If \(x\) is squared, it opens up or down. If \(y\) is squared, it opens left or right!

B. Ellipses (The "Squashed Circles")

Standard Form: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
Example: If you see \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), the graph stretches 3 units left/right (because \(\sqrt{9}=3\)) and 2 units up/down (because \(\sqrt{4}=2\)).

C. Hyperbolas (The "Two-Part" Curves)

These look like two mirrored curves that never touch certain lines called asymptotes.
Standard Form: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) (opens left/right) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\) (opens up/down).
Common Mistake: Don't mix up the minus sign! The variable with the positive sign is the direction the "bowls" open.

D. Rational Functions (Fractions with \(x\))

You will often see \(y = \frac{ax+b}{cx+d}\) or \(y = \frac{ax^2+bx+c}{dx+e}\).
The most important thing here is finding the Asymptotes—the "invisible electric fences" the graph approaches but never crosses.

Quick Review: Finding Asymptotes
1. Vertical Asymptote: Set the denominator to zero and solve for \(x\).
2. Horizontal/Oblique Asymptote: Use Long Division to rewrite the fraction. The non-fractional part is your asymptote equation!

Section Summary:

Every graph has "ID features": Symmetry (is it a mirror image?), Intersects (where it hits the axes), and Turning Points (peaks and valleys). Always look for these first!

2. The Art of Transformation

Think of transformations as "filters" or "modifiers" applied to a base graph \(y = f(x)\). There is a simple trick to remember how they work: Outside vs. Inside.

"Outside" Changes (Affects \(y\)) - Very Logical

If the change is outside the bracket, it affects the vertical direction (up/down). These do exactly what they look like!
- \(y = f(x) + a\): Shifts the graph UP by \(a\) units.
- \(y = af(x)\): Stretches the graph vertically by factor \(a\). (If \(a\) is negative, it reflects over the x-axis).

"Inside" Changes (Affects \(x\)) - Counter-Intuitive

If the change is inside the bracket, it affects the horizontal direction (left/right). These are "backwards" and often trip students up!
- \(y = f(x + a)\): Shifts the graph LEFT by \(a\) units. (Yes, plus means left!)
- \(y = f(ax)\): Stretches the graph horizontally by factor \(\frac{1}{a}\). (If \(a=2\), the graph gets twice as narrow!)

Memory Aid: "Inside is Opposite"
Whenever you see a change inside the bracket with \(x\), think: "This does the opposite of what I expect." Adding moves it negative, multiplying divides the width.

The Order Matters!

If you have multiple transformations, the standard safe order is SRT:
1. Stretch
2. Reflection
3. Translation (Shifting)

Key Takeaway:

Always identify the "parent" function first, then apply transformations step-by-step. Don't try to do them all at once!

3. Specialized Graph Relationships

The syllabus requires you to relate three special types of graphs to the original \(y = f(x)\).

A. The Modulus: \(y = |f(x)|\)

The Rule: No negative \(y\) values allowed.
How to draw: Keep everything above the x-axis as it is. For any part below the x-axis, reflect it upward to make it positive.

B. The Modulus: \(y = f(|x|)\)

The Rule: The function only sees positive \(x\) values.
How to draw: Ignore/erase everything on the left side (negative \(x\)). Take the right side and mirror it onto the left side. It will always be perfectly symmetrical across the y-axis.

C. The Reciprocal: \(y = \frac{1}{f(x)}\)

This is the trickiest one, but follow these steps:
1. Where \(f(x) = 0\), the new graph has a Vertical Asymptote.
2. Where \(f(x)\) is very large, \(\frac{1}{f(x)}\) is very small (approaches zero).
3. Where \(f(x)\) has a maximum, \(\frac{1}{f(x)}\) has a minimum (and vice versa).
4. The points where \(y = 1\) or \(y = -1\) stay exactly where they are (Invariant points).

Did you know?

The Reciprocal transformation is like a seesaw. When one goes up, the other goes down!

4. Simple Parametric Equations

Usually, we see \(y\) as a function of \(x\). In parametric equations, both \(x\) and \(y\) are controlled by a third "hidden" variable, usually called \(t\) or \(\theta\) (the parameter).
Example: \(x = 2t\), \(y = t^2\).

How to handle them:
1. Substitution: Try to make \(t\) the subject in the \(x\) equation and plug it into the \(y\) equation to get a normal "Cartesian" equation.
2. GC is your friend: Learn how to switch your Graphing Calculator to "Parametric Mode" to see what these look like! It's perfectly legal and encouraged in the syllabus.

Section Summary:

Parametric equations are just another way to describe a path. Instead of \(y\) directly depending on \(x\), they both follow the lead of \(t\).

Final Tips for Success

1. Use your Graphing Calculator (GC): The syllabus explicitly includes "use of a graphing calculator." If you are stuck on a shape, plot it! Use the GC to check your asymptotes and turning points.
2. Label everything: In exams, you lose easy marks if you don't label your asymptotes with their equations (e.g., \(x = 2\)) and your intercepts with coordinates (e.g., \((0, 5)\)).
3. Practice the "Standard Moves": Transformations are a guaranteed topic. Master the "Inside/Outside" rule and you'll have a huge advantage.

Don't worry if you find reciprocal graphs or oblique asymptotes hard—they are the most advanced part of this chapter. Keep practicing the standard parabolas and transformations first, and the rest will fall into place!