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Functions y = ax², y = ax² + n, y = a(x − m)²

Mathematics Grade 9 Difficulty 2/5

Beginning of the lesson

Quadratic Functions: Transformations of Parabolas

Big Idea: Every quadratic function creates a U-shaped curve called a parabola. By changing the values of $a$, $n$, and $m$, we can stretch, flip, shift up/down, and shift left/right any parabola. Mastering these transformations lets you graph any quadratic by sight!

Part 1: The Parent Function $y = x^2$

Before we explore transformations, we need a starting point. The simplest quadratic function is:

$$y = x^2$$

This is called the parent function. Its key features are:

  • Vertex: $(0, 0)$ — the lowest point
  • Axis of symmetry: $x = 0$ (the y-axis)
  • Opens: Upward
  • Domain: All real numbers
  • Range: $y \geq 0$
Interactive visualization — available in the full lesson

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In the full lesson

  • The complete explanation
  • Worked examples
  • Common mistakes and how to avoid them
  • Test questions: 132
  • AI tutor that answers questions about the topic
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