Quadratic Functions: Properties, Vertex Form, and Graphing Techniques

Properties of a Parabola

Definition and Standard Form

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

• $f(x) = ax^2 + bx + c$

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

• a, b, and c can be any real numbers, with $a \neq 0$

The graph of a quadratic function is called a parabola.

Key Features of Parabolas

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

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

• 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: Square Function

• For $f(x) = x^2$:

  • Vertex: $(0, 0)$ (minimum)

  • Axis of Symmetry: $x = 0$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

  • Increasing on $(0, \infty)$, decreasing on $(-\infty, 0)$

Vertex Form & Transformations

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is:

• $f(x) = a(x - h)^2 + k$

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

• If $a > 0$, parabola opens upward (minimum); if $a < 0$, opens downward (maximum)

Transformations

• Horizontal shift: $h$ units right if $h > 0$, left if $h < 0$

• Vertical shift: $k$ units up if $k > 0$, down if $k < 0$

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

• Reflection: If $a < 0$, reflects over the x-axis

Steps to Graph from Vertex Form

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

2. Draw 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 points 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: $a(x^2 + \frac{b}{a}x) + c$

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

3. Rewrite as a perfect square trinomial: $a(x - h)^2 + k$

Example

• Given $f(x) = x^2 + 6x + 7$

• Rewrite: $f(x) = (x + 3)^2 - 2$

• Vertex: $(-3, -2)$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use
Factoring	When the quadratic has obvious factors or $c = 0$
Square Root Property	When the equation is of the form $x^2 = k$ or $a(x - h)^2 = k$
Complete the Square	When the leading coefficient is 1 and $b$ is even
Quadratic Formula	When factoring is difficult or unsure which method to use

Practice and Application

Identifying Key Features from Graphs

• Given a graph, identify the vertex, axis of symmetry, and whether the vertex is a minimum or maximum.

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

Graphing Quadratic Functions

1. Write the function in vertex form if possible.

2. Identify the vertex $(h, k)$ and axis of symmetry $x = h$.

3. Find x-intercepts (solve $f(x) = 0$) and y-intercept ($f(0)$).

4. Determine the domain (always $(-\infty, \infty)$) and range (depends on vertex and direction).

5. Identify intervals where the function is increasing or decreasing.

6. Plot points and sketch the parabola.

Example: Graphing from Vertex Form

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

• Vertex: $(-1, 2)$ (maximum)

• Axis of Symmetry: $x = -1$

• Opens downward ($a < 0$)

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, 2]$

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

Example: Graphing from Standard Form

• Given $f(x) = -2x^2 - 4x + 6$

• Convert to vertex form by completing the square

• Identify vertex, axis of symmetry, intercepts, domain, range, and intervals of increase/decrease

Summary

• Quadratic functions are fundamental in algebra and are characterized by their parabolic graphs.

• Key features include vertex, axis of symmetry, intercepts, domain, and range.

• Vertex form makes it easy to graph and identify transformations.

• Completing the square is a useful technique for converting standard form to vertex form.

• Practice identifying and graphing these features to master quadratic functions.