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All courses Algebra I · F-BF.3 18 of 76
Analyze graph transformations caused by f(x)+k, kf(x), f(kx), and f(x+k); identify even/odd functions.

Describe a vertical shift of a function

Problem
Describe the vertical shift from \(f(x)\) to \(f(x)+4\).
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Hint

Notice that the +4 is added after f(x), so it changes the output values rather than the input values.

Adding a number outside f(x) changes the output values. A positive outside number raises every y-value by that amount.

Solution walkthrough

01

Locate the added value

\[g(x)~=~f(x)~+~4\]

The 4 is outside f's input, so it is added after the original function produces an output.

02

Track a general point

\[(x,~y)~\text{on}~f~->~(x,~y~+~4)~\text{on}~g\]

Every x-coordinate stays fixed while every y-coordinate increases by 4.

03

Connect coordinates to movement

\[y~+~4~\text{is}~4~\text{units}~\text{above}~y\]

Increasing all outputs by the same positive amount translates the entire graph vertically upward without changing its shape.

04

State the transformation

\[\text{shift}~\text{up}~4~\text{units}\]

The exact vertical shift from f(x) to f(x) + 4 is up 4 units.

+

Another way

  1. Choose any sample output y; the transformed output y + 4 is four units higher at the same x-coordinate.

!

Common mistake

A plus sign outside the function changes outputs, so it is not a rightward input shift or a vertical scale.

Solution walkthrough video