Quadratic Functions: Properties, Vertex Form, and Transformations

Quadratic Functions and Parabolas

Definition and Standard Form

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

• $f(x) = ax^2 + bx + c$

• Where a, b, and c are real numbers, and a \neq 0.

Examples:

• $f(x) = x^2$

• $f(x) = 2x^2 + 3x - 7$

• $f(x) = x^2 + 1$

The graph of a quadratic function is called a parabola.

Key Properties of Parabolas

• Vertex: The highest or lowest point on the parabola (minimum or maximum).

• Axis of Symmetry: A vertical line that divides the parabola into two mirror images. For $f(x) = ax^2 + bx + c$, the axis is $x = -\frac{b}{2a}$.

• x-intercepts: Points where the graph crosses the x-axis (solutions to $f(x) = 0$).

• y-intercept: Point where the graph crosses the y-axis ($f(0)$).

• Domain: All real numbers, $(-\infty, \infty)$.

• Range: Depends on whether the parabola opens up or down.

Example Table: Properties of $f(x) = x^2$ and $f(x) = -(x+2)^2 + 1$

Function	Vertex	Axis of Symmetry	Opens	Range
$f(x) = x^2$	(0, 0)	$x = 0$	Up	$[0, \infty)$
$f(x) = -(x+2)^2 + 1$	(-2, 1)	$x = -2$	Down	$(-\infty, 1]$

Vertex Form and Transformations

Vertex Form

The vertex form of a quadratic function is:

• $f(x) = a(x-h)^2 + k$

• Where $(h, k)$ is the vertex of the parabola.

• a determines the direction and width of the parabola.

Transformations:

• Horizontal shift: $h$ units left/right

• Vertical shift: $k$ units up/down

• Reflection: If $a < 0$, the parabola opens downward.

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

Example: $f(x) = 2(x-1)^2 + 3$ has vertex at $(1, 3)$, opens upward, and is vertically stretched.

Graphing Quadratic Functions in Vertex Form

1. Identify the vertex $(h, k)$.

2. Find the axis of symmetry ($x = h$).

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

4. Find y-intercept by computing $f(0)$.

5. Plot the vertex and intercepts, then sketch the curve.

Standard Form to Vertex Form: Completing the Square

Completing the Square

To convert $f(x) = ax^2 + bx + c$ to vertex form:

1. Factor out $a$ from the $x^2$ and $x$ terms.

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

3. Move the subtraction outside the parentheses.

4. Simplify to get $f(x) = a(x-h)^2 + k$.

Example:

• Given $f(x) = x^2 + 6x + 7$

• Rewrite: $f(x) = (x^2 + 6x) + 7$

• Add and subtract $(\frac{6}{2})^2 = 9$ inside: $f(x) = (x^2 + 6x + 9) - 9 + 7$

• Vertex form: $f(x) = (x + 3)^2 - 2$

Practice and Application

Identifying Key Features from Graphs

• Locate the vertex and determine if it is a minimum or maximum.

• Find the axis of symmetry from the graph.

• Determine intercepts and intervals of increase/decrease.

Example: For $f(x) = -2x^2 - 4x + 6$, complete the square to find vertex form, then graph and identify all key features.

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Notes
Factoring	Obvious factors, $c = 0$	Quick if applicable
Square Root Property	Form $x^2 = k$	Use when no $x$ term
Completing the Square	Leading coefficient is 1 or even	Useful for vertex form
Quadratic Formula	Can't easily factor	General method

Additional info:

• All quadratic functions have domain $(-\infty, \infty)$.

• The range depends on the vertex and direction of opening.

• Intervals of increase/decrease are determined by the vertex and the direction the parabola opens.