Quadratic Functions: Properties, Vertex Form, and Graphing Techniques

Quadratic Functions

Properties of a Parabola

A quadratic function is a polynomial of degree 2, typically written in the standard form:

• $f(x) = ax^2 + bx + c$

• Examples: $f(x) = 2x^2 + 3x - 7$, $f(x) = x^2 + x + 1$

All quadratic functions graph as parabolas. The coefficient a determines whether the parabola opens upward ($a > 0$) or downward ($a < 0$).

• Vertex: The highest or lowest point of the parabola, given by $x = -\frac{b}{2a}$

• Axis of Symmetry: The vertical line passing through the vertex, $x = -\frac{b}{2a}$

• Domain: $(-\infty, \infty)$

• Range: $[\text{vertex } y, \infty)$ if $a > 0$; $(-\infty, \text{vertex } y]$ if $a < 0$

• Increasing/Decreasing Intervals: The function decreases to the vertex and increases after (for $a > 0$), and vice versa for $a < 0$.

Example: For $f(x) = x^2 - 2x + 1$, the vertex is at $(1, 0)$, axis of symmetry $x = 1$, and the parabola opens upward.

Vertex Form & Transformations

The vertex form of a quadratic function is useful for identifying transformations and the vertex directly:

• $f(x) = a(x - h)^2 + k$

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

• Vertical stretch/compression: $|a| > 1$ stretches, $0 < |a| < 1$ compresses

• Horizontal shift: $h$ units

• Vertical shift: $k$ units

Example: $f(x) = 2(x - 3)^2 + 4$ has vertex $(3, 4)$, opens upward, and is vertically stretched.

Graphing Quadratic Functions

To graph a quadratic function, follow these steps:

1. Find the vertex using $x = -\frac{b}{2a}$ (standard form) or $(h, k)$ (vertex form).

2. Determine the axis of symmetry.

3. Find the y-intercept by evaluating $f(0)$.

4. Find x-intercepts by solving $f(x) = 0$.

5. Identify the domain and range.

6. Determine intervals of increase and decrease.

7. Sketch the graph using the above information.

Example: For $f(x) = x^2 + 1$, vertex is $(0, 1)$, axis of symmetry $x = 0$, y-intercept $1$, opens upward.

Standard Form to Vertex Form: Completing the Square

To convert a quadratic from standard form to vertex form, use completing the square:

• Given $f(x) = ax^2 + bx + c$

• Factor out $a$ from the first two terms

• Add and subtract $(\frac{b}{2a})^2$ inside the bracket

• Rewrite as $a(x - h)^2 + k$

Example: $f(x) = x^2 + 4x + 5$ Complete the square: $f(x) = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1$

Summary Table: Quadratic Function Properties

Form	Vertex	Axis of Symmetry	Opens	Domain	Range
Standard: $ax^2 + bx + c$	$(-\frac{b}{2a}, f(-\frac{b}{2a}))$	$x = -\frac{b}{2a}$	$a > 0$ up, $a < 0$ down	$(-\infty, \infty)$	$[\text{vertex } y, \infty)$ or $(-\infty, \text{vertex } y]$
Vertex: $a(x-h)^2 + k$	$(h, k)$	$x = h$	$a > 0$ up, $a < 0$ down	$(-\infty, \infty)$	$[k, \infty)$ or $(-\infty, k]$

Practice Problems

• Identify the vertex and axis of symmetry for given quadratic graphs.

• State whether the vertex is a minimum or maximum.

• Graph quadratic functions and label key features: vertex, axis of symmetry, intercepts, domain, range, and intervals of increase/decrease.

Additional info: These notes cover the essential properties and graphing techniques for quadratic functions, including transformations and completing the square, as required for Precalculus students.