Understanding Vertical Shifts

Vertical shifts are an important way to move graphs up or down without changing how they look. These shifts help us see how functions change when we make different adjustments.

What Are Vertical Shifts?

Vertical shifts happen when we add or subtract a number from a function.

For any function, like $f(x)$, we can show vertical shifts like this:

• Upward Shift: This is when we add a number, so it looks like $f(x) + k$ (where $k$ is greater than 0).
• Downward Shift: This is when we subtract a number, so it looks like $f(x) - k$ (where $k$ is also greater than 0).

How Do They Affect the Graph?

1. Positioning:

   • Upward Shift: When we add $k$, every point on the graph moves up by $k$ units. For example, if $f(x) = x^2$, changing it to $g(x) = x^2 + 3$ moves the whole graph up by 3 units.
   • Downward Shift: When we subtract $k$, every point moves down by $k$ units. So, if $g(x) = x^2 - 2$, the graph shifts down by 2 units.

2. Impact on Key Points:

   • If the starting point of the function was at $(h, k)$, after a vertical shift, the new point will be at $(h, k + m)$, where $m$ is how much we shifted it.

3. Function Values:

   • The output values of the function change directly. For example, if $f(2) = 4$, then:
     • With an upward shift of 3, $g(2) = f(2) + 3 = 7$.
     • With a downward shift of 2, $g(2) = f(2) - 2 = 2$.

Quick Recap

Vertical shifts move the graph straight up or down but keep its shape the same. Knowing how these shifts work helps us graph functions better and solve problems in algebra.