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Parent Functions & Transformations · Core principles

Read shifts, reflections, and scale

Understand each part of a transformed function.

Updated 10/13/2026Permanent link

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The idea
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The idea

The transformation rule

For nonzero $a$ and $b$ , use

y=a\,f\bigl(b(x-h)\bigr)+k.

$h$ moves the graph right if positive, left if negative.
$k$ moves it up if positive, down if negative.
$a<0$ reflects it across the $x$ -axis before the vertical shift. The vertical scale factor is $\lvert a\rvert$ .
$b<0$ reflects it horizontally before the horizontal shift. The horizontal scale factor is $\frac1{\lvert b\rvert}$ .

Factors greater than 1 stretch; factors between 0 and 1 compress. An inside multiplier of 2 therefore gives a horizontal compression by $\frac12$ .

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Example 1

$y=-2(x-3)^2+4$ starts with $f(x)=x^2$ : reflect across the $x$ -axis, stretch vertically by 2, shift right 3, then up 4. The vertex is $(3,4)$ .

Notice and plan
Pause and identify the mathematical feature or graph clue before following the calculation.
Work it through
Start with y equals x squared. Reflect across the x axis and stretch vertically by two. Shift three units right, then four units up. The result is y equals negative two times the quantity x minus three, squared, plus four. The vertex is three, four.
Check and understand
The parent point one, one becomes four, two: move its input right three and multiply its output by negative two before adding four. This verifies the order of the moves.

Parent (dashed)
Final graph

Follow the reasoning, then check the result.

Read the narration · spoken math transcript

Example 1. Pause and identify the mathematical feature or graph clue before following the calculation. Start with y equals x squared. Reflect across the x axis and stretch vertically by two. Shift three units right, then four units up. The result is y equals negative two times the quantity x minus three, squared, plus four. The vertex is three, four. The parent point one, one becomes four, two: move its input right three and multiply its output by negative two before adding four. This verifies the order of the moves.

Common mistake

Watch the grouping

Factor an inside expression before reading its shift: $2x-6=2(x-3)$ , so the shift is right 3, not right 6.

A useful point rule is $(u,v)\mapsto(h+\frac ub,\;k+av)$ . This avoids ambiguity when several transformations happen together.

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Explore the idea

Try it yourself

Move a parabola

Predict how each slider will move or reshape the curve.

Vertical scale1Horizontal shift0Vertical shift0

y=1(x-(0))^2+(0),\quad (h,k)=(0,0)

The vertex is (0, 0). The parabola opens upward; its vertical scale is 1.

Challenge: Put the vertex at (2, 3) and make the parabola open downward.

Used in practice

Horizontal ShiftsVertical ShiftsReflectionsStretches and CompressionsWrite the Equation

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Recognize the parent function