Understanding Vertical Shifts in Linear Equations

Vertical shifts change where linear equations are located on a graph. They change the starting point on the y-axis, but the steepness of the line stays the same.

A linear equation is often written like this: $y = mx + b$ Here, $m$ is the slope (how steep the line is), and $b$ is the y-intercept (where the line crosses the y-axis).

What Are Vertical Shifts?

1. Moving Upward:

   • When we add a number, $k$, to the equation, it changes to: $y = mx + (b + k)$
   • This means the whole line moves up by $k$ units.
   • For example, if $k = 3$ and the original equation is $y = 2x + 1$, the new equation would be $y = 2x + 4$.

2. Moving Downward:

   • If we subtract a number, $k$, from the equation, the new equation looks like: $y = mx + (b - k)$
   • This moves the entire line down by $k$ units.
   • For instance, if we subtract $2$ from the original equation, it becomes $y = 2x - 1$.

Important Points About Vertical Shifts

• Slope: The slope ($m$) does not change. This means the steepness of the line stays the same, no matter if we shift it up or down.
• Y-Intercept: The y-intercept ($b$) does change. When we shift upward, $b$ goes up. When we shift downward, $b$ goes down.

How It Looks on a Graph

When you draw these changes, the lines will be parallel. This means they run next to each other without crossing:

• If the original line starts at (0, 1) and has a slope of 2, moving it up by 3 means it will now start at (0, 4).
• If you move it down by 2, it will start at (0, -1).

Why This Matters

Knowing about vertical shifts helps us understand real-life situations, like in economics and science, where we use equations to show relationships. For example, if a company’s revenue goes up by a fixed cost, a vertical shift shows how the basic revenue changes without affecting how much profit they make (the slope).

Final Thoughts

In short, vertical shifts change the position of linear equations on a graph but keep their slope the same. Understanding these shifts is important for studying linear equations and helps us see how these equations respond to different changes.