Lesson 1.4

Vertex Form

The ultimate graphing shortcut. By combining , , and , we can sketch any quadratic function in seconds.

Introduction

We've stretched and shifted our parabolas separately. Now, we combine these transformations into a single, powerful equation known as Vertex Form, which instantly reveals the graph's most important features.

Past Knowledge

You've learned that stretches, shifts left/right, and shifts up/down.

Today's Goal

We combine these into Vertex Form: .

Future Success

This form is arguably more useful than Standard Form for graphing because the vertex is visible instantly.

Key Concepts

Vertex Form Equation

V
The Vertex (h, k)This is the turning point. Flip the sign of h, keep k the same.
A
Axis of Symmetry ($x = h$)The vertical line passing through the vertex.
a
Shape and DirectionSame rules as before: Positive = Up, Negative = Down. Large = Narrow, Fraction = Wide.

Standard parabola shifted Right 2 and Up 1.

Worked Examples

Example 1: Identifying Attributes

Basic

Identify the vertex and axis of symmetry for .

Identify h (Flip the Sign)

Equation has . Think . So, .

Identify k (Keep the Sign)

Equation ends with . So, .

Vertex: (-3, -4)
Axis of Sym: x = -3

Example 2: Graphing

Intermediate

Graph .

Plot the Vertex

. Plot point (1, 3).

Use the 1-3-5 Step Pattern

Since , from the vertex, move Over 1, Up 1. Then Over 1, Up 3.

Example 3: Writing the Equation

Advanced

Write the equation of a parabola with Vertex passing through .

Plug in Vertex (h, k)

Find 'a' using Point (x, y)

Plug in for x and y:

Equation:

Common Pitfalls

Forgetting to Flip 'h'

In , the vertex is at -5, not 5. In , the vertex is at 2, not -2. Always think: "Inside is Opposite".

Confusing Vertex Form with Standard Form

Vertex form has the parenthesis squared . Standard form is . You can't just pick "b" out of vertex form.

Real-Life Applications

Trajectory Physics: In games like Angry Birds, the flight path is pre-calculated using vertex form. The developers pick a starting point, a peak height (vertex), and a landing target, and the physics engine fits a parabola to that path.