Compressions and Stretches in Vertex Form
In vertex form f(x) = a(x-h)² + k, the coefficient a controls vertical stretches and compressions of the parabola — a key transformation topic in enVision Algebra 1 Chapter 10 for Grade 11. When |a| > 1, the parabola stretches vertically (narrower). When 0 < |a| < 1, it compresses (wider). When a < 0, the parabola reflects across the x-axis. For f(x) = 3(x-1)² + 2, the graph is stretched vertically by factor 3 (narrower than y = x²). For f(x) = ½(x+2)² - 3, it is compressed by factor ½ (wider). For f(x) = -2(x-4)² + 1, it is stretched and reflected downward.
Key Concepts
How to identify and apply vertical stretches and compressions using vertex form.
In the vertex form $f(x) = a(x h)^2 + k$, the coefficient $a$ controls vertical stretches and compressions: If $|a| 1$: vertical stretch by factor of $|a|$ If $0 < |a| < 1$: vertical compression by factor of $|a|$ If $a < 0$: reflection across x axis (plus stretch/compression).
Common Questions
How does the value of a in f(x) = a(x-h)² + k affect the parabola shape?
If |a| > 1: vertically stretched (narrower). If 0 < |a| < 1: vertically compressed (wider). If a < 0: reflected across the x-axis (opens downward). The magnitude |a| determines width; the sign determines direction.
Is f(x) = 3(x-1)² + 2 narrower or wider than y = x²?
Narrower. The coefficient |a| = 3 > 1 means a vertical stretch by factor 3, making the parabola narrow.
Is f(x) = ½(x+2)² - 3 narrower or wider than y = x²?
Wider. The coefficient |a| = ½ < 1 means a vertical compression, making the parabola wider and flatter.
What does a negative a value do to the parabola?
It reflects the parabola across the x-axis so it opens downward. For f(x) = -2(x-4)² + 1, the parabola opens downward and is also stretched vertically by factor 2.
Does the a coefficient affect the vertex location?
No. The vertex is still (h, k) regardless of a. The coefficient a only affects the shape (width and orientation), not the vertex position.