MCR3U Parent Functions: Notes and Practice

Before you can describe a transformed function like $y = 2 x - 1 + 3$ , you need to know what the untransformed version looks like. A parent function is the simplest form of a family of functions — no stretches, no shifts, nothing added. Learn the shape, domain, range and a couple of key points for each parent function, and every transformation becomes "start from this picture, then move it." It is the same idea as learning the basic shapes before drawing: a transformed graph is never drawn from scratch, it is a parent graph that has been moved, flipped or stretched.

Fig 1The five parent functions side by side: linear, quadratic, square root, reciprocal, absolute value.

The five parent functions

Function	Name	Domain	Range	Key points
$y = x$	Linear	$x \in R$	$y \in R$	(-1,-1), (0,0), (1,1)
$y = x^{2}$	Quadratic	$x \in R$	$y \geq 0$	(-2,4), (-1,1), (0,0), (1,1), (2,4)
$y = x$	Square root	$x \geq 0$	$y \geq 0$	(0,0), (1,1), (4,2), (9,3)
$y = \frac{1}{x}$	Reciprocal	$x \neq = 0$	$y \neq = 0$	(-1,-1), (1,1), (2, 0.5), (0.5, 2); asymptotes at x=0, y=0
$y = ∣ x ∣$	Absolute value	$x \in R$	$y \geq 0$	(-2,2), (-1,1), (0,0), (1,1), (2,2)

Notice the shapes: $y = x$ is a straight diagonal line. $y = x^{2}$ is a symmetric "U" (a parabola) with its lowest point (vertex) at the origin. $y = x$ starts at the origin and curves up and to the right, but only exists for $x \geq 0$ since you can't take the square root of a negative number and stay real. $y = \frac{1}{x}$ has two separate branches that hug the axes but never touch them — the axes are asymptotes. $y = ∣ x ∣$ looks like a "V" with its corner at the origin.

Why the reciprocal graph never touches an axis

x \to 0

\frac{1}{x}

grows without bound (positive or negative), so the graph shoots off vertically near

x = 0

but never actually reaches it. As

x \to \pm \infty

\frac{1}{x} \to 0

, so the graph flattens toward the

x

-axis but never touches it either.

VideoA quick tour of the common parent graphs with their domains and ranges. It covers ten functions; for this lesson focus on the linear, quadratic, square root, reciprocal and absolute value ones.Watch on YouTube

Why each shape looks the way it does

Each shape follows from what the rule does to a number. Squaring makes negatives positive and treats $x$ and $- x$ identically, so $y = x^{2}$ is symmetric about the $y$ -axis and never goes below $0$ . Between $- 1$ and $1$ it stays closer to the $x$ -axis than $y = x$ does (because $0. 5^{2} = 0.25$ is smaller than $0.5$ ), and outside that interval it climbs far above it (because $3^{2} = 9$ ). The square root undoes squaring for non-negative numbers, so $y = x$ is exactly the right half of the parabola $y = x^{2}$ reflected in the line $y = x$ : the points $(1, 1), (4, 2), (9, 3)$ are the parabola's points $(1, 1), (2, 4), (3, 9)$ with coordinates swapped. That is why it rises quickly at first and then flattens. The reciprocal turns small numbers into big ones and big numbers into small ones ( $\frac{1}{0.1} = 10$ , $\frac{1}{10} = 0.1$ ), which produces the two asymptotes. The absolute value measures distance from $0$ , so it keeps positive inputs and flips negative ones:

∣ x ∣ = {x, - x, x \geq 0 x < 0

Two straight half-lines with slopes 1 and −1 meeting at the origin — that is why the graph is a V with a sharp corner.

Recognizing a parent function from a table or graph

Given only a table of values, look at the pattern of outputs. Constant first differences signal linear; constant second differences signal a quadratic ( $y = x^{2}$ -type); outputs restricted to $x \geq 0$ with slow growth signal a square root; outputs that shrink toward zero as $x$ grows (and blow up near $x = 0$ ) signal a reciprocal; a symmetric table whose values change by the same amount on each side of the minimum signals absolute value. Compare the five parent functions on the same inputs:

x	y = x	y = x²	y = √x	y = 1/x	y = \|x\|
−2	−2	4	undefined	−0.5	2
−1	−1	1	undefined	−1	1
0	0	0	0	undefined	0
1	1	1	1	1	1
4	4	16	2	0.25	4

The same inputs give very different output patterns — that pattern is each function's fingerprint.

$y = x$ and $y = x^{2}$ are both defined for every real $x$ — the difference is the shape (line vs. curve) and the range.
$y = x$ and $y = 1/ x$ both have domain restrictions, but for different reasons (a radicand vs. a denominator).
$y = x^{2}$ and $y = ∣ x ∣$ have the same domain and range, but different shapes (smooth curve vs. sharp corner) and different growth: at $x = 4$ the parabola is at $16$ but the V is only at $4$ .
All five pass through $(1, 1)$ — so that point never tells them apart. Use a negative input or a large input instead.

From a parent to a family

Every function you meet in the next lessons is a parent function with changes applied. $y = 2 (x - 3)^{2} + 1$ belongs to the quadratic family, $y = - x + 4$ to the square-root family, and $y = \frac{1}{x - 2} + 5$ to the reciprocal family. To graph one, you take the parent's key points from the table above and move each one with a mapping rule, which is why those key points are worth memorizing now. The parent's domain and range also transform in predictable ways: shifting $y = x$ four units left moves its starting point from $(0, 0)$ to $(- 4, 0)$ , so the domain becomes $x \geq - 4$ . Quadratic relations get extra attention later (forms, completing the square, solving) because they model so many real situations.

Sketching a parent function quickly

Plot three to five key points, not dozens. For

y = x^{2}

use

x = - 2, - 1, 0, 1, 2

; for

y = x

use the perfect squares

0, 1, 4, 9

; for

y = \frac{1}{x}

use

\pm \frac{1}{2}, \pm 1, \pm 2

and draw the two asymptotes first as dashed lines. Then join the points with the correct shape: a smooth U, a half-arch, two branches, or a sharp V.

Parent Functions

Key things to remember

The five parent functions

Why each shape looks the way it does

Recognizing a parent function from a table or graph

From a parent to a family

Worked examples

Where marks get lost

Flashcards 9

Practice