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All courses Math II · F-BF.3 16 of 73
Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.

Interpret \(f(x)+k\) as a vertical shift of a quadratic or absolute-value function

Problem
Identify the vertical shift from parent function f(x)=x² to transformed function g(x)=x²+5.
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Hint

Compare \(g(x)=x^2+5\) to the parent \(f(x)=x^2\) and focus on the \(+5\) outside the square.

Adding a positive number outside the function shifts the graph up by that amount.

Solution walkthrough

01

Identify the parent function

\[f(x)~=~x^2\]

This is the original parabola before any transformation.

02

Write the transformed function

\[g(x)~=~x^2+5\]

The transformed function keeps the same \(x^2\) shape and adds \(5\) outside.

03

Interpret the outside addition

\[+5\]

Adding \(5\) to every output moves every y-value up by \(5\).

04

State the vertical shift

\[\text{shift}:~\text{up}~5\]

So the graph of \(x^2\) is shifted upward \(5\) units.

+

Another way

  1. Check a point: the parent vertex moves from \((0,0)\) to \((0,5)\), confirming an upward shift of \(5\).

!

Common mistake

Treating the \(+5\) as a horizontal change, even though it is added outside the function and changes the outputs.

Answer graph showing the exact transformed function and the requested shift, scale, reflection, comparison, or contextual feature.
shift: up 5