Shifting Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Shifting Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Shifting a function translates its graph horizontally or vertically without changing its shape: f(x - h) + k shifts right by h and up by k.

Shifting is like sliding the entire graph on the coordinate plane — the function's shape is completely unchanged, only its position moves.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Horizontal shifts are 'backwards': f(x - h) shifts RIGHT h units.

Common stuck point: Inside the function = horizontal (opposite sign). Outside = vertical (same sign).

Sense of Study hint: Ask: where does the new function equal what the old function did at x = 0? That tells you the direction and size of the shift.

Worked Examples

Example 1

easy
Starting from f(x)=x^2, describe and sketch the transformations for g(x)=(x-3)^2+2.

Solution

  1. 1
    Horizontal shift: x-3 inside the function shifts the parabola 3 units to the right. Vertex moves from (0,0) to (3,0).
  2. 2
    Vertical shift: +2 outside shifts the graph 2 units up. Vertex moves from (3,0) to (3,2).
  3. 3
    Result: upward-opening parabola with vertex at (3,2); same shape as f(x)=x^2.

Answer

Right 3, up 2; vertex at (3, 2)
Horizontal shifts come from changes inside the function argument: f(x-h) shifts right by h (left if h<0). Vertical shifts come from adding a constant outside: f(x)+k shifts up by k. The vertex formula is (h,k).

Example 2

medium
The graph of f(x)=e^x is shifted left 2 units and down 5 units to give g(x). Write g(x), find g(0), and determine if the horizontal asymptote changes.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
The point (4, 7) is on the graph of y=f(x). Find the corresponding point on each shifted graph: (a) y=f(x-1)+3, (b) y=f(x+5)-2.

Example 2

medium
Write the equation of the function whose graph is y=\sqrt{x} shifted right 9 and reflected over the x-axis. State the domain.
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Related Concepts

Function Transformation

Background Knowledge

These ideas may be useful before you work through the harder examples.

transformation