The Transformation Gallery

At a glance

Solo · launched mid-unit, due at the unit’s close · a gallery walk on the due date · eight graphs, eight captions, one artist’s statement

What you are making

Choose one parent function — $f(x)=x^2$, $\sqrt{x}$, or $\frac{1}{x}$ — and curate its family. Your gallery holds eight portraits: the parent, then seven children chosen so that every move in $y=af(k(x-d))+c$ appears somewhere, including at least one portrait that combines three moves. Each portrait carries a caption: the equation, the moves in the order you applied them, the domain and range, and one point tracked from parent to image.

You finish with a short artist’s statement: which portrait was hardest to hang, and what it taught you about the recipe.

Milestones

• Parent chosen; its key points tabled and its domain and range stated
• Four single-move portraits sketched by hand, then confirmed with sliders in Using Desmos
• Three combined-move portraits, including one using $a$, $k$, and $d$ together
• Captions written — order of moves stated and defended, one point tracked through each portrait
• Artist’s statement drafted; gallery assembled for the walk

Notice that the milestones above are a checkbox list — tick them off on paper as you go. The order matters: hands before technology, per Transformations of Functions.

How it is assessed

This is evidence of growth, not a race — How Marks Work explains the difference. During the gallery walk you will answer one question about a portrait of your teacher’s choosing, and your answer counts as much as the graph does. Afterwards, your Math Journal entry on the hardest portrait becomes part of the record.

Success criteria

Quality	What it looks like in your gallery
Complete family	Every letter of the recipe appears at least once
Honest sketches	Hand-drawn first; technology confirms, not creates
Tracked points	One parent point followed through every portrait
Precise captions	Domain, range, and order of moves stated exactly
A real statement	The artist’s statement names a genuine difficulty

If a portrait will not behave

The two inside letters act backwards — $f(2x)$ squeezes and $f(x-2)$ slides right. If a sketch and Desmos disagree, track one point by hand and see which of the two is lying. It is never Desmos.

Curriculum connection

A1.8

determine, through investigation using technology, the roles of the parameters $a$, $k$, $d$, and $c$ in functions of the form $y=af(k(x-d))+c$, and describe these roles in terms of transformations on the graphs of $f(x)=x$, $f(x)=x^2$, $f(x)=\sqrt{x}$, and $f(x)=\frac{1}{x}$ (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes) Sample problem: Investigate the graph $f(x)=3(x-d{)}^2+5$ for various values of $d$, using technology, and describe the effects of changing $d$ in terms of a transformation.

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A1.9

sketch graphs of $y=af(k(x-d))+c$ by applying one or more transformations to the graphs of $f(x)=x$, $f(x)=x^2$, $f(x)=\sqrt{x}$, and $f(x)=\frac{1}{x}$, and state the domain and range of the transformed functions Sample problem: Transform the graph of $f(x)$ to sketch $g(x)$, and state the domain and range of each function, for the following: $f(x)=\sqrt{x}$, $g(x)=\sqrt{x-4}$; $f(x)=\frac{1}{x}$, $g(x)=-\frac{1}{x+1}$.

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A1.3

explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions $f(x)=x$, $f(x)=x^2$, $f(x)=\sqrt{x}$, and $f(x)=\frac{1}{x}$; describe the domain and range of a function appropriately (e.g., for $y=x^2+1$, the domain is the set of real numbers, and the range is $y\ge 1$); and explain any restrictions on the domain and range in contexts arising from real-world applications Sample problem: A quadratic function represents the relationship between the height of a ball and the time elapsed since the ball was thrown. What physical factors will restrict the domain and range of the quadratic function?

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