Transformations of Functions

At the boards, your group graphed $y=\sqrt{x}$, $y=\sqrt{x}+3$, $y=\sqrt{x-2}$, and $y=-\sqrt{x}$ — and someone said “these are all the same graph, just moved.” That sentence is the whole topic. In Grade 10 you learned three moves that turn $y=x^2$ into any parabola. The news this year is better than a new set of rules: there are no new rules. The same moves work on every function you will ever meet.

The recipe

Start from any parent function $y=f(x)$ — squaring, square root, $\frac{1}{x}$, and soon exponentials and sinusoids. Then

$y=af(k(x-d))+c$

is the parent after four moves:

Letter	The move	Watch for
$a$	Stretch vertically by $a$; negative flips over the x-axis	Heights multiply
$k$	Compress horizontally by $k$; negative flips over the y-axis	Widths divide by $k$
$d$	Slide right by $d$	Hides inside the bracket with a minus sign
$c$	Slide up by $c$	The only honest letter of the four

The two inside letters, $k$ and $d$, act on $x$ before the function does — which is why they behave backwards: $f(x-2)$ slides right, not left, and $f(2x)$ squeezes rather than stretches. The function is a machine; whatever you do to the input, the graph shows the reverse.

One point at a time

A transformation is really an instruction for moving points: the point $(x,y)$ on the parent lands at $\left(\frac{x}{k}+d,ay+c\right)$ on the image. To sketch $y=-2f\left(\frac{1}{2}(x+1)\right)+3$, take three or four known points of the parent and push each one through:

• Multiply each $x$ by 2 (undoing the $\frac{1}{2}$), then subtract 1.
• Multiply each $y$ by $-2$, then add 3.
• Plot the image points; join with the parent’s shape in mind.
• Confirm one point in the equation — or the whole curve in Using Desmos.

Stretch first, slide second — the same order as Grade 10, for the same reason: sliding first drags your anchor points out of position, and the graph betrays it immediately.

Where this pays off: every function family in this course — exponential, sinusoidal, and the ones you meet after it — arrives as a parent plus this recipe. Learn the recipe once and each new family costs you one parent graph, not a chapter. Transformations Practice turns it from a procedure into a reflex.

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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