Lesson 1.1

The Parent Function

Meet the parabola—the fundamental shape of gravity, antennas, and architecture. Everything starts with the simple equation .

Introduction

Quadratic functions are the mathematical backbone of gravity, projectile motion, and satellite dishes. Before we can model these complex real-world phenomena, we must understand the simplest quadratic of them all: the parent function.

Past Knowledge

You are an expert at graphing lines like . Lines have a constant rate of change (slope) and are straight.

Today's Goal

We explore the simplest non-linear function: the Quadratic Parent Function, . We analyze its U-shape, vertex, and symmetry.

Future Success

This U-shape (the Parabola) models everything from projectile motion to satellite dishes. Mastering the parent is key to mastering all quadratics.

Key Concepts

The Quadratic Parent Function

The simplest quadratic function is . Its graph is a symmetrical curve called a Parabola.

1Vertex: The turning point where the graph changes direction. For , the vertex is at the origin .
2Axis of Symmetry: The vertical line that splits the parabola in half. Here, it is the y-axis, .
3Domain: . You can square ANY real number.
4Range: . Squaring a real number never produces a negative result.

The classic "U" shape of the parabola.

Worked Examples

Example 1: Graphing by Table

Basic

Create a table of values and graph .

Choose x-values around the vertex (0)

x	y = x²
-2	(-2)² = 4
-1	(-1)² = 1
0	0
1	(1)² = 1
2	(2)² = 4

Example 2: Domain and Range

Concept

Determine the Domain and Range of .

Domain (Input)

Can we square a negative number? Yes. Zero? Yes. Positive? Yes.
The graph extends infinitely to the left and right.

Range (Output)

Look at the graph. Does it ever go below the x-axis? No. The lowest y-value is 0.

Example 3: Symmetry in Action

Analysis

Verify that for .

Evaluate Positive x

Evaluate Negative x

Since both outputs are 9, the points and are mirror images across the y-axis.

Common Pitfalls

Squaring Negatives Incorrectly

Remember that but . When substituting a negative number into , always use parentheses: .

Confusing Domain and Range

The Domain covers ALL x-values (left to right), but the Range only covers positive y-values (bottom to top).

Real-Life Applications

Antennas and Reflectors: Satellite dishes are shaped like parabolas because of a special property: any signal hitting the dish reflects directly to the focus (a point inside the curve). This allows the dish to collect weak signals from space and concentrate them into a strong signal for your TV or internet.

Projectile Motion: Any object thrown under gravity (a ball, a jumping dolphin, water from a fountain) follows a parabolic path.