Vertical Shifting (Translation) of Graphs

This interactive tool helps you explore and understand the vertical shifting (translation) of the graph of a function when a constant \(k\) is added to the function. The function to be analyzed is of the form \(f(x) + k\).

Select Function

Vertical Shift: \(k = \) 0.0

-5 0.0 5

Function Information

Original Function: \(f(x) = x^2\)

\(f(x) + k = x^2 + 0.0\)

When \(k > 0\): Graph shifts upward by \(k\) units
When \(k < 0\): Graph shifts downward by \(|k|\) units
When \(k = 0\): No vertical shift

Original Function \(f(x)\)

Shifted Function \(f(x) + k\)

Graph Visualization

Tutorial

How does adding a constant \(k\) to a function affect the graph of this function?

Click on any function button above to select it.
Use the slider to set the constant \(k\) to different values and observe the effect on the graph.
Answer the following questions:

Question 1: What happens to the graph when \(k\) is positive?

Question 2: What happens to the graph when \(k\) is negative?

Question 3: How does the vertical shift affect the range of the function?

Question 4: What happens when \(k = 0\)? Does the graph change?

Explain analytically: For a function \(f(x)\), the transformed function \(f(x) + k\):

Shifts the graph upward by \(k\) units if \(k > 0\)
Shifts the graph downward by \(|k|\) units if \(k < 0\)
Leaves the graph unchanged if \(k = 0\)
Changes the range of the function by adding \(k\) to all output values
Does not change the domain of the function

Key observations:

Vertical shifting moves the entire graph up or down without changing its shape
The slope/steepness of the graph remains unchanged
All points \((x, y)\) on the original graph become \((x, y + k)\) on the shifted graph
X-intercepts may change or disappear with vertical shifting