Lesson 2-5: Graphing Parabolas — MathPath

Core Concept

Key Features of a Parabola

Vertex — the turning point: either a minimum (a > 0) or maximum (a < 0)
Axis of Symmetry — the vertical line through the vertex: $x = -b/(2a)$
y-intercept — where the graph hits the y-axis: $(0, c)$
x-intercepts (roots) — where the graph hits the x-axis: solve $ax²+bx+c=0$
Opens up/down — upward if a > 0; downward if a < 0

Example

Sketch f(x) = x² − 4x + 3

1. Direction: a = 1 > 0 \to opens upward (\cup shape) 2. Axis of symmetry: x = -(-4)/(2\cdot1) = 4/2 = 2 3. Vertex: f(2) = 4 - 8 + 3 = -1 \to vertex: (2, -1) 4. y-intercept: c = 3 \to point (0, 3) 5. x-intercepts: x² - 4x + 3 = 0 \to (x-1)(x-3) = 0 \to x=1, x=3

Quick Method

Sketching From Vertex Form

If you already have vertex form $f(x) = a(x-h)² + k$ , sketching is immediate:

Vertex: $(h, k)$
Axis of symmetry: $x = h$
Opens up if a > 0, down if a < 0
Wider if |a| < 1, narrower if |a| > 1

Your Turn

Try It Yourself

Q: For $f(x) = x² + 2x - 8$ , find: (a) the vertex, (b) axis of symmetry, (c) x-intercepts.

Show Answer

(a) Axis: x = -2/(2\cdot1) = -1. Vertex: f(-1) = 1 - 2 - 8 = -9 \to (-1, -9) (b) Axis of symmetry: x = -1 (c) x² + 2x - 8 = 0 \to (x+4)(x-2) = 0 \to x = -4 or x = 2

Key Takeaways

1-Minute Summary

Vertex x-coordinate: $x = -b/(2a)$
a > 0 → ∪ shape (minimum); a < 0 → ∩ shape (maximum)
y-intercept is always at (0, c)
x-intercepts: solve the quadratic = 0

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