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 of quadratic functions:

• $f(x) = x^2$

• $f(x) = 2x^2 + 3x - 7$

• $f(x) = (x + 2)^2 + 1$

The graph of a quadratic function is called a parabola.

Key Properties of Parabolas

• Vertex: The highest or lowest point on the parabola, depending on whether it opens upward (minimum) or downward (maximum).

• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetric halves. Its equation is $x = h$ for vertex $(h, k)$.

• Intercepts: Points where the parabola crosses the axes.

  • x-intercepts: Solve $f(x) = 0$.

  • y-intercept: Compute $f(0)$.

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

• Range: Depends on the direction the parabola opens.

  • If $a > 0$, range is $[k, \infty)$ (minimum at vertex).

  • If $a < 0$, range is $(-\infty, k]$ (maximum at vertex).

• Intervals of Increase/Decrease:

  • Decreasing on $(-\infty, h)$

  • Increasing on $(h, \infty)$

Example Table: Properties of Quadratic Functions

Function	Vertex	Axis of Symmetry	x-intercepts	y-intercept	Domain	Range
$f(x) = x^2$	(0, 0) [MIN]	$x = 0$	0	0	$(-\infty, \infty)$	$[0, \infty)$
$f(x) = (x + 2)^2 + 1$	(-2, 1) [MIN]	$x = -2$	None (if discriminant < 0)	$f(0) = 5$	$(-\infty, \infty)$	$[1, \infty)$

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:

  • If $a > 0$, opens upward (minimum).

  • If $a < 0$, opens downward (maximum).

  • $|a| > 1$ causes vertical stretch; $0 < |a| < 1$ causes vertical compression.

Transformations

• Horizontal shift: $h$ shifts the graph left/right.

• Vertical shift: $k$ shifts the graph up/down.

• Reflection: Negative $a$ reflects the graph over the x-axis.

Example:

• $f(x) = 2(x - 1)^2 - 4$ has vertex at $(1, -4)$, opens upward, and is vertically stretched.

Steps to Graph a Quadratic Function in Vertex Form

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

2. Draw the axis of symmetry $x = h$.

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

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

5. Plot the vertex, intercepts, and sketch the parabola.

Standard Form to Vertex Form: Completing the Square

Converting Standard Form to Vertex Form

To rewrite $f(x) = ax^2 + bx + c$ in vertex form, complete the square:

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: $f(x) = (x + 3)^2 - 2$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Notes
Factoring	When quadratic factors easily	Set $c = 0$
Square Root Property	When in form $x^2 = k$	Take $\pm \sqrt{k}$
Complete the Square	When leading coefficient is 1 or even	Useful for vertex form
Quadratic Formula	When factoring is difficult	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Practice and Application

Identifying Properties from Graphs

• Given a graph, locate the vertex and determine if it is a minimum or maximum.

• Find the axis of symmetry by identifying the vertical line through the vertex.

• Determine intercepts by observing where the graph crosses the axes.

• State the domain and range based on the direction the parabola opens.

• Identify intervals of increase and decrease.

Example: Graphing and Analyzing Quadratic Functions

• Given $f(x) = \frac{1}{2}(x + 1)^2 + 2$:

  • Vertex: $(-1, 2)$ [MIN]

  • Axis of Symmetry: $x = -1$

  • Domain: $(-\infty, \infty)$

  • Range: $[2, \infty)$

  • Increasing on $(-1, \infty)$, Decreasing on $(-\infty, -1)$

Practice Problems

• Convert $f(x) = 3x^2 + 12x$ to vertex form by completing the square.

• Graph $f(x) = -2x^2 - 4x + 6$ and identify all key properties.

• Given a graph, state the vertex, axis of symmetry, and intervals of increase/decrease.

Additional info: These notes cover the essential properties and graphing techniques for quadratic functions, including vertex form, transformations, and completing the square, as required in College Algebra (Chapter 3: Quadratic Functions and Equations).