Lesson 5.13

Vertical Shifts

What happens when you add a number to the end of a function? The whole graph takes an elevator ride.

Introduction

We know that in is the y-intercept. But in the world of functions, we call this a Vertical Shift because it slides the entire line up or down without changing its angle.

Past Knowledge

Lesson 5.7 (Slope-Intercept Form). You already know that changing changes where the line starts.

Today's Goal

Describe the transformation as a shift up or down.

Future Success

This "k" logic applies to EVERYTHING: Parabolas (), Absolute Value (), and Trig ().

Key Concepts

The "k" Shift

If k is Positive ()

Shift UP by units.

Example: moves up 3.

If k is Negative ()

Shift DOWN by units.

Example: moves down 5.

Notice: The lines are Parallel. The slope never changes.

Worked Examples

Example 1: Describing Shifts

Basic

Describe the transformation from to .

Step 1: Compare

The part is identical. The slope hasn't changed.

The only difference is the at the end.

Step 2: Interpret k

. Negative means Down.

Vertical shift down 4 units.

Example 2: Writing Equations

Intermediate

The parent function is . Write the equation for a line shifted UP 2 units.

Apply $+k$

"Up 2" means add 2 to the end.

Example 3: Multiple Shifts?

Advanced

Let . Let be the result of shifting DOWN 5 units. Find .

Method

We take the WHOLE and subtract 5.

g(x) = (3x + 1) - 5

Simplify

Combine the constants ().

Note: It ended at -4 because it started at +1 and went down 5.

Common Pitfalls

Adding to x instead of y

Vertical shifts happen outside the function: . If you do , that's a Horizontal shift (left/right), which is much weirder. Keep the number at the very end.

Forgetting the original b

If and you shift down 2, the new eq is NOT . It is . You must combine it with the existing intercept.

Real-Life Applications

Salary Raises: If everyone in a company gets a \$5,000 raise, the entire salary graph shifts UP by 5000. Use the same slope (pay scale), just higher starting point.