Introduction to Graphs and Transformations

In Mathematics, we often use equations to describe relationships. But sometimes, looking at a list of numbers or symbols doesn't tell the whole story. Graphs allow us to "see" the math. By understanding how to sketch curves and how to move them around—called transformations—you can solve complex problems visually and algebraically.

In this chapter, we focus on three main types of graphs: linear, quadratic, and cubic. We will also learn how to slide, stretch, and flip these graphs using simple rules.

1. Standard Graphs You Must Know

Before we can transform graphs, we need to know what the "parent" or original versions look like. In the P1 syllabus, you are only required to know three types:

Linear Graphs

These are straight lines. The standard form is \(y = mx + c\).

  • \(m\) is the gradient (how steep it is).
  • \(c\) is the \(y\)-intercept (where it crosses the vertical axis).

Quadratic Graphs

These form a curve called a parabola. The standard form is \(y = ax^2 + bx + c\).

  • If \(a\) is positive, the graph is "u-shaped" (a happy face).
  • If \(a\) is negative, the graph is "n-shaped" (a sad face).
  • Vertex: The lowest or highest point of the curve.
  • Line of Symmetry: Every quadratic is symmetrical about a vertical line passing through its vertex.

Cubic Graphs

These involve an \(x^3\) term. They usually have an "S" shape.

  • A positive \(x^3\) starts low and ends high (bottom-left to top-right).
  • A negative \(x^3\) starts high and ends low (top-left to bottom-right).
  • They can have up to two "turning points" (bumps).

Quick Tip: For more detail on solving these equations algebraically, see the Algebra and polynomials chapter. For now, focus on their shapes!

2. Geometrical Interpretation of Solutions

When we solve two equations simultaneously, we are actually finding the points of intersection—the exact coordinates where the two graphs cross each other.

  • If you solve a linear and quadratic equation and get two distinct real roots, the line crosses the curve at two points.
  • If you get equal roots (the discriminant is zero), the line is a tangent—it touches the curve at exactly one point.
  • If there are no real roots, the line and curve never meet.

3. The Four Rules of Transformation

Imagine you have a basic graph \(y = f(x)\). A transformation changes the position or shape of that graph. In the AS Level exam, you will only be tested on one transformation at a time.

Type 1: Vertical Translation — \(y = f(x) + a\)

This "slides" the graph up or down.

  • What happens: Every \(y\)-coordinate increases by \(a\).
  • Movement: If \(a\) is positive, it moves up. If \(a\) is negative, it moves down.
  • Vector notation: This is a translation by the vector \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).

Type 2: Horizontal Translation — \(y = f(x + a)\)

This "slides" the graph left or right. Be careful! This one is often the opposite of what you expect.

  • What happens: The graph moves horizontally.
  • Movement: If you see \((x + 2)\), the graph moves left by \(2\). If you see \((x - 2)\), the graph moves right by \(2\).
  • Vector notation: This is a translation by the vector \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\).

Type 3: Vertical Stretch — \(y = a f(x)\)

This pulls the graph away from the \(x\)-axis or squashes it toward it.

  • What happens: Every \(y\)-coordinate is multiplied by \(a\).
  • Scale Factor: The scale factor is \(a\) in the \(y\)-direction.
  • Reflection: If \(a = -1\), the graph is reflected in the \(x\)-axis (it flips upside down).

Type 4: Horizontal Stretch — \(y = f(ax)\)

This squashes or pulls the graph horizontally. Like the horizontal translation, this is "counter-intuitive."

  • What happens: Every \(x\)-coordinate is divided by \(a\).
  • Scale Factor: The scale factor is \(\frac{1}{a}\) in the \(x\)-direction.
  • Example: \(y = f(2x)\) actually makes the graph half as wide.
  • Reflection: If \(a = -1\), the graph is reflected in the \(y\)-axis (it flips left-to-right).

4. Summary Table for Quick Revision

Don't worry if these seem tricky at first! Use this table to remember the "Inside/Outside" rule.

Transformation Where is \(a\)? Effect on Coordinates Description
\(f(x) + a\) Outside Add \(a\) to \(y\) Vertical Translation
\(a f(x)\) Outside Multiply \(y\) by \(a\) Vertical Stretch (SF \(a\))
\(f(x + a)\) Inside Subtract \(a\) from \(x\) Horizontal Translation
\(f(ax)\) Inside Divide \(x\) by \(a\) Horizontal Stretch (SF \(\frac{1}{a}\))

Memory Aid:
Outside = Obvious (Up is +, Down is -, Stretch is \(\times a\)).
Inside = Inverse (Right is -, Left is +, Stretch is \(\div a\)).

5. Common Mistakes to Avoid

  • Mixing up \(x\) and \(y\): Remember that if the change is "inside" the bracket with \(x\), it affects the horizontal \(x\)-axis. If it's "outside" the bracket, it affects the vertical \(y\)-axis.
  • Forgetting the \(x\)-stretch fraction: For \(y = f(3x)\), students often say "stretch factor 3." It is actually stretch factor \(\frac{1}{3}\).
  • Translation Signs: Remember that \(f(x + 5)\) moves the graph in the negative \(x\) direction (left).

Key Takeaways

  • Linear, Quadratic, and Cubic: Know their basic shapes before you start.
  • Intersections: Solving equations tells you where graphs cross.
  • One at a time: In Unit P1, you only need to handle a single transformation (no combining stretches and translations yet!).
  • Translations: Moving the graph without changing its shape.
  • Stretches: Changing the width or height of the graph.