Quadratic Functions: Graphs, Vertices, and Properties

Quadratic Functions and Their Graphs

Introduction

Quadratic functions are a fundamental topic in College Algebra, typically expressed in the form f(x) = ax^2 + bx + c. Their graphs are parabolas, and understanding their properties is essential for analyzing equations and inequalities.

Graph Orientation

• Upward or Downward Opening: The direction in which a parabola opens depends on the sign of the leading coefficient a in f(x) = ax^2 + bx + c.

• If a > 0, the parabola opens upward.

• If a < 0, the parabola opens downward.

• Example: For f(x) = -5(x - 1)^2 + 12, the coefficient of the squared term is -5, so the parabola opens downward.

Vertex of a Parabola

• Definition: The vertex is the highest or lowest point on the graph of a quadratic function.

• For a function in vertex form f(x) = a(x - h)^2 + k, the vertex is at (h, k).

• Example: For f(x) = -5(x - 1)^2 + 12, the vertex is at (1, 12).

Axis of Symmetry

• Definition: The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves.

• For f(x) = a(x - h)^2 + k, the axis of symmetry is x = h.

• Example: For f(x) = -5(x - 1)^2 + 12, the axis of symmetry is x = 1.

Maximum and Minimum Values

• Maximum Value: If the parabola opens downward (a < 0), the vertex represents the maximum value of the function.

• Minimum Value: If the parabola opens upward (a > 0), the vertex represents the minimum value.

• Example: For f(x) = -5(x - 1)^2 + 12, the maximum value is 12 at x = 1.

Domain and Range

• Domain: The set of all possible input values (x) for the function. For any quadratic function, the domain is all real numbers (−∞ < x < ∞).

• Range: The set of all possible output values (f(x)). For a downward-opening parabola, the range is f(x) ≤ k, where k is the y-coordinate of the vertex.

• Example: For f(x) = -5(x - 1)^2 + 12, the range is f(x) ≤ 12.

Summary Table: Properties of Quadratic Functions

Property	General Form	Example: f(x) = -5(x - 1)^2 + 12
Direction	Upward if a > 0, Downward if a < 0	Downward (a = -5)
Vertex	(h, k)	(1, 12)
Axis of Symmetry	x = h	x = 1
Maximum/Minimum	Maximum if a < 0, Minimum if a > 0	Maximum at y = 12
Domain	All real numbers	All real numbers
Range	Upward: y ≥ k; Downward: y ≤ k	y ≤ 12

Key Formulas

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x - h)^2 + k$

• Axis of Symmetry: $x = -\frac{b}{2a}$ (for standard form)

• Vertex: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

Example Problem

• Given: $f(x) = -5(x - 1)^2 + 12$

• (a) Direction: Downward (since $a = -5$)

• (b) Vertex: $(1, 12)$

• (c) Axis of Symmetry: $x = 1$

• (d) Maximum Value: $12$ at $x = 1$

• (e) Domain: All real numbers

• (f) Range: $f(x) \leq 12$