Lesson 1.3

Transformations: Shifting

We can move the entire parabola anywhere on the grid without changing its shape. It's time to slide left, right, up, and down.

Introduction

Parabolas aren't stuck at the origin. We can slide them horizontally and vertically to position them anywhere on the coordinate plane, allowing us to model motion starting from any location.

Past Knowledge

You've graphed at the origin (0,0). You've also stretched it. Now we need to move the vertex itself.

Today's Goal

We learn how adding constants inside or outside the square affects the position. We introduce (horizontal) and (vertical).

Future Success

Identifying and is the core skill for Vertex Form and solving equations by completing the square.

Key Concepts

Vertical Shifts (The 'k' value)

Adding a number outside the square moves the graph Up or Down.

• If , shift UP.
• If , shift DOWN.

Horizontal Shifts (The 'h' value)

Adding/Subtracting a number inside the square moves the graph Left or Right. WARNING: It's the opposite of what you think!

• moves RIGHT 3 (because h=3).
• moves LEFT 3 (because h=-3).

Visualizing shifts from the origin.

Worked Examples

Example 1: Vertical Shift

Basic

Graph .

Identify the Shift

The +2 is outside the square. This is a vertical shift UP by 2 units.

Move the Vertex

Old Vertex: (0,0).
New Vertex: (0, 0+2) = (0, 2).

Example 2: Horizontal Shift

Concept

Graph .

Identify 'h'

Inside the parentheses we see . Think: "What value of x makes the inside zero?"
. So, we shift RIGHT 4.

Vertex Position

The new vertex is at (4, 0).

Example 3: Combined Shifts

Advanced

Identify the vertex of .

Horizontal Shift (Inside)

means . Shift LEFT 2.

Vertical Shift (Outside)

means . Shift DOWN 5.

New Vertex

Vertex:

Common Pitfalls

The "Inside Logic" Trap

Seeing makes your brain want to go to negative 3 on the number line. Fight this instinct! "Inside is Opposite". Minus moves Right. Plus moves Left.

Real-Life Applications

Video Game Physics: In 2D platformers (like Mario), jumping follows a parabolic arc. When the character moves forward while jumping, the game calculates a shifted parabola. If Mario jumps from position x=100 and reaches a peak height of 50 units, the code essentially uses .