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Identify horizontal translations of functions

A horizontal translation changes the input of a function while preserving all output values: in y=f(x−h)y=f(x-h), h>0h>0 shifts the graph hh units right, whereas h<0h<0 shifts it ∣h∣|h| units left. The corresponding point mapping is (x,y)↦(x+h,y)(x,y)\mapsto(x+h,y), so the domain shifts while the range remains unchanged; interpreting this relationship across equations, graphs, and tables helps prevent reversing the sign inside the function. The focus is on single-constant translations, not compositions of transformations or more abstract parameterized settings.

Detailed Explanation: Identify horizontal translations of functions

A horizontal translation changes the input of a function, so the graph moves left or right while its output values stay the same.

In the form

y=f(x−h),y=f(x-h),
  • if h>0h>0, the graph shifts hh units right;
  • if h<0h<0, the graph shifts ∣h∣\vert h \vert units left.

The sign can feel backward: an expression such as x−3x-3 means a shift right 3, not left 3.

Example: Let

f(x)=xf(x)=\sqrt{x}

and define

g(x)=f(x−3).g(x)=f(x-3).

Describe the translation, point mapping, domain, and range of gg.

Step 1: Identify the value of hh.

Compare g(x)=f(x−3)g(x)=f(x-3) with f(x−h)f(x-h). Here,

h=3.h=3.

Since hh is positive, the graph shifts 3 units right.

Step 2: Apply the point mapping.

A point (x,y)(x,y) on the original graph moves according to

(x,y)↦(x+h,y).(x,y)\mapsto(x+h,y).

Therefore,

(x,y)↦(x+3,y).(x,y)\mapsto(x+3,y).

For example, the point (0,0)(0,0) on y=xy=\sqrt{x} moves to (3,0)(3,0) on y=g(x)y=g(x).

Step 3: Write the new function and domain.

Because

g(x)=f(x−3)=x−3,g(x)=f(x-3)=\sqrt{x-3},

the expression inside the square root must be nonnegative:

x−3≥0,x-3\ge 0,

so

x≥3.x\ge 3.

The domain is

[3,∞).[3,\infty).

Step 4: Identify the range.

A horizontal translation does not change the output values. The original function f(x)=xf(x)=\sqrt{x} has range

[0,∞),[0,\infty),

so gg has the same range:

[0,∞).[0,\infty).

Thus, g(x)=f(x−3)g(x)=f(x-3) is the graph of ff shifted 3 units right. Its domain shifts from [0,∞)[0,\infty) to [3,∞)[3,\infty), while its range remains [0,∞)[0,\infty).

Learn by doing: Identify horizontal translations of functions

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