Vertical Shift
Key Formula
g(x)=f(x)+k
Where:
- f(x) = The original function
- g(x) = The transformed function after the vertical shift
- k = The number of units the graph shifts vertically. If k > 0, the graph moves up; if k < 0, the graph moves down.
Worked Example
Problem: Given the function f(x) = x², describe and graph the vertical shift that produces g(x) = x² + 3.
Step 1: Identify the original function and the transformation. The original function is f(x) = x². The new function adds 3 to every output.
g(x)=f(x)+3=x2+3
Step 2: Determine the direction and magnitude of the shift. Since k = 3 is positive, the entire graph moves up by 3 units.
Step 3: Verify with a specific point. On f(x), when x = 2, f(2) = 4, giving the point (2, 4). On g(x), g(2) = 4 + 3 = 7, giving the point (2, 7). The point moved up exactly 3 units.
f(2)=4⟶g(2)=4+3=7
Answer: The graph of g(x) = x² + 3 is the parabola f(x) = x² shifted up 3 units. The vertex moves from (0, 0) to (0, 3), and every other point rises by 3 as well.
Another Example
Problem: Start with h(x) = |x| and apply a vertical shift of −4. Write the new function and describe the transformation.
Step 1: Apply the vertical shift formula with k = −4.
g(x)=∣x∣+(−4)=∣x∣−4
Step 2: Since k is negative, every point on the V-shaped graph of |x| moves down 4 units. The vertex moves from (0, 0) to (0, −4).
Step 3: Check: when x = 3, h(3) = 3 and g(3) = 3 − 4 = −1. The point (3, 3) becomes (3, −1), confirming a downward shift of 4.
g(3)=∣3∣−4=−1
Answer: The new function is g(x) = |x| − 4, which is the absolute value graph shifted down 4 units.
Frequently Asked Questions
How do you tell the difference between a vertical shift and a horizontal shift?
A vertical shift adds or subtracts a constant outside the function, changing the y-values: g(x) = f(x) + k. A horizontal shift adds or subtracts a constant inside the function's input, changing the x-values: g(x) = f(x − h). Vertical shifts move the graph up or down; horizontal shifts move it left or right.
Does a vertical shift change the shape of the graph?
No. A vertical shift is a rigid transformation — it moves every point the same distance in the same direction. The shape, width, and orientation of the graph stay exactly the same. Only the position changes.
Vertical Shift vs. Horizontal Shift
A vertical shift modifies the output of a function (g(x) = f(x) + k), sliding the graph up or down. A horizontal shift modifies the input (g(x) = f(x − h)), sliding the graph left or right. Both are rigid transformations that preserve the graph's shape. A common source of confusion is that horizontal shifts work in the opposite direction to the sign: f(x − 3) shifts right, not left. Vertical shifts, by contrast, are more intuitive — adding a positive number shifts up, and subtracting shifts down.
Why It Matters
Vertical shifts appear constantly when modeling real situations. For example, if a baseline temperature function gives average monthly temperatures, adding a constant represents a climate that is uniformly warmer. Understanding vertical shifts also builds the foundation for combining multiple transformations — shifts, stretches, and reflections — to quickly sketch or interpret any transformed function.
Common Mistakes
Mistake: Confusing the sign of k: thinking g(x) = f(x) − 5 shifts the graph up.
Correction: Subtracting from the output always moves the graph down. Here k = −5, so every point drops 5 units. A positive k shifts up; a negative k shifts down.
Mistake: Mixing up vertical and horizontal shifts by placing the constant inside the function argument.
Correction: A vertical shift changes the output: f(x) + k. A horizontal shift changes the input: f(x − h). The constant's position — outside vs. inside — determines the direction of the shift.
Related Terms
- Horizontal Shift — Shifts the graph left or right instead
- Shift — General term for any translation of a graph
- Transformation — Broader category including shifts, stretches, reflections
- Vertical — Direction along the y-axis
- Plane Figure — A flat shape that can be shifted vertically
- Translation — Another name for a shift transformation
- Reflection — Flips a graph, often combined with shifts