Shifting Functions Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Starting from f(x)=x2f(x)=x^2, describe and sketch the transformations for g(x)=(x−3)2+2g(x)=(x-3)^2+2.

Solution

  1. 1
    Horizontal shift: x−3x-3 inside the function shifts the parabola 33 units to the right. Vertex moves from (0,0)(0,0) to (3,0)(3,0).
  2. 2
    Vertical shift: +2+2 outside shifts the graph 22 units up. Vertex moves from (3,0)(3,0) to (3,2)(3,2).
  3. 3
    Result: upward-opening parabola with vertex at (3,2)(3,2); same shape as f(x)=x2f(x)=x^2.

Answer

Right 33, up 22; vertex at (3,2)(3, 2)
Horizontal shifts come from changes inside the function argument: f(x−h)f(x-h) shifts right by hh (left if h<0h<0). Vertical shifts come from adding a constant outside: f(x)+kf(x)+k shifts up by kk. The vertex formula is (h,k)(h,k).

About Shifting Functions

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

Learn more about Shifting Functions →

More Shifting Functions Examples

Example 2 medium

The graph of [formula] is shifted left [formula] units and down [formula] units to give [formula]. W

Example 3 easy

The point [formula] is on the graph of [formula]. Find the corresponding point on each shifted graph

Example 4 medium

Write the equation of the function whose graph is [formula] shifted right [formula] and reflected ov

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Related Concepts

Function Transformation