2.2.4 Transformations of Quadratic Functions

Apply translations, reflections, and stretches to quadratic functions and understand how parameters affect the parabola.

定义

Transformations of quadratic functions involve applying translations, reflections, and stretches to the parent function \(f(x) = x^2\). These transformations modify the position, orientation, and shape of the parabola on the coordinate plane. A general transformed quadratic function can be written in vertex form as \(f(x) = a(x-h)^2 + k\), where: - \(a\) determines the vertical stretch/compression and reflection across the x-axis - \(h\) represents the horizontal translation (shift left/right) - \(k\) represents the vertical translation (shift up/down) - The vertex of the parabola is located at the point \((h, k)\) \nTransformations can also be expressed in the form \(f(x) = a(x-h)^2 + k\) where \(|a| > 1\) produces a vertical stretch, \(0 < |a| < 1\) produces a vertical compression, and \(a < 0\) reflects the parabola across the x-axis. Horizontal stretches and compressions are represented by replacing \(x\) with \(\frac{x}{b}\) in the function, where \(|b| > 1\) produces a horizontal stretch and \(0 < |b| < 1\) produces a horizontal compression.

核心公式

• \(f(x) = a(x-h)^2 + k\)
• \(f(x) = ax^2 + bx + c\) (standard form)
• \(h = -\frac{b}{2a}\) (x-coordinate of vertex)
• \(k = f(h) = a(h)^2 + b(h) + c\) (y-coordinate of vertex)
• \(f(x) = a(x - r_1)(x - r_2)\) (factored form with roots \(r_1\) and \(r_2\))

易错点

• ⚠️ Confusing the direction of horizontal translations: replacing \(x\) with \((x+h)\) shifts the graph LEFT by \(h\) units, not right. Students often incorrectly think \((x+h)\) shifts right.
• ⚠️ Incorrectly applying the vertical stretch factor: when \(a > 1\), the parabola stretches vertically (becomes narrower), but students sometimes think it compresses or fail to recognize how it affects the y-values.
• ⚠️ Mixing up the order of transformations or applying them incorrectly when converting between vertex form and standard form, especially when completing the square.
• ⚠️ Forgetting that a negative value of \(a\) reflects the parabola across the x-axis, causing the parabola to open downward, and then miscalculating the vertex or range of the function.