Function Transformations - Precalculus & Trigonometry Step by Step

Transform y = x² into y = 2(x-3)² + 1 by applying vertical stretch, horizontal shift, and vertical shift, then map key points.

Key point mapping: base point (bx, by) maps to (bx + h, A·by + k). Example: y = 2(x - 3)² + 1. A = 2, h = 3, k = 1. (0, 0) → (3, 1) and (1, 1) → (4, 3).

Example

Describe shifts and stretches from the parent function using exact values.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

y= 2 (x- 3 )^{ 2 }+ 1 \quad\text{from}\quad y= x^{2}

Step 2 — Vertical stretch

Apply a vertical stretch by factor 2.

\text{vertical stretch } \hl{2}

Step 3 — Horizontal shift

Shift the parent graph right 3.

\text{right } \hl{3}

Step 4 — Vertical shift

Shift the graph up 1.

\text{up } \hl{1}

Step 5 — Map first point

Map point (0, 0) to (3, 1).

( 0 , 0 )\to( \hl{3} , \hl{1} )

Step 6 — Map second point

Map point (1, 1) to (4, 3).

( 1 , 1 )\to( \hl{4} , \hl{3} )

function-transformations Starting from a parent function y = x², the transformed form y = A(x - h)² + k applies three changes: Vertical stretch by A (multiply all y-values by A). Horizontal shift right by h (replace x with x - h). Vertical shift up by k (add k to all y-values).