Parent Functions

A parent function is the simplest member of a family of functions. In this course, you’ll build lots of graphs by shifting, stretching, and flipping four parent functions, so knowing their shapes and a few key points by heart makes sketching much faster.

Key ideas

The four parent functions

The orange dots are the key points listed in the table.

Function	Name	Key points	Domain	Range
$f(x) = x$	linear	$(-2, -2)$, $(-1, -1)$, $(0, 0)$, $(1, 1)$, $(2, 2)$	$\{x \in \mathbb{R}\}$	$\{y \in \mathbb{R}\}$
$f(x) = x^2$	quadratic	$(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, $(2, 4)$	$\{x \in \mathbb{R}\}$	$\{y \in \mathbb{R} \mid y \ge 0\}$
$f(x) = \sqrt{x}$	square root	$(0, 0)$, $(1, 1)$, $(4, 2)$, $(9, 3)$	$\{x \in \mathbb{R} \mid x \ge 0\}$	$\{y \in \mathbb{R} \mid y \ge 0\}$
$f(x) = \dfrac{1}{x}$	reciprocal	$(-2, -\tfrac{1}{2})$, $(-1, -1)$, $(-\tfrac{1}{2}, -2)$, $(\tfrac{1}{2}, 2)$, $(1, 1)$, $(2, \tfrac{1}{2})$	$\{x \in \mathbb{R} \mid x \ne 0\}$	$\{y \in \mathbb{R} \mid y \ne 0\}$

What to remember about each shape

• $y = x$ is a straight line through the origin with slope $1$.
• $y = x^2$ is a U-shaped parabola with its vertex at $(0, 0)$. It’s symmetric about the $y$-axis, so $x$ and $-x$ give the same output.
• $y = \sqrt{x}$ starts at $(0, 0)$ and only goes to the right, rising more and more slowly. It’s the top half of a sideways parabola. For $x \ge 0$, it undoes $x^2$: see inverse functions.
• $y = \dfrac{1}{x}$ has two separate pieces. It gets closer and closer to the $x$-axis and the $y$-axis but never touches them.

Asymptotes

An asymptote is a line that a graph gets closer and closer to without ever reaching. $y = \dfrac{1}{x}$ has two:

• a vertical asymptote at $x = 0$, because you can’t divide by zero
• a horizontal asymptote at $y = 0$, because $\dfrac{1}{x}$ is never $0$, but gets very close when $x$ is large

Worked examples

Example 1: Recognizing a parent function from a table

Which parent function matches each table?

(a)

$x$	$0$	$1$	$4$	$9$
$y$	$0$	$1$	$2$	$3$

(b)

$x$	$1$	$2$	$4$
$y$	$1$	$0.5$	$0.25$

Solution.

(a) Each output is the square root of the input: $\sqrt{9} = 3$. This is $y = \sqrt{x}$.

(b) Each output is $1$ divided by the input: $\dfrac{1}{4} = 0.25$. This is $y = \dfrac{1}{x}$.

Example 2: Comparing x² and √x

Compare $x^2$ and $\sqrt{x}$ at $x = \dfrac{1}{4}$ and at $x = 4$. Where are they equal?

Solution.

At $x = \tfrac{1}{4}$: $\left(\tfrac{1}{4}\right)^2 = \tfrac{1}{16}$ and $\sqrt{\tfrac{1}{4}} = \tfrac{1}{2}$. Here $\sqrt{x}$ is bigger.

At $x = 4$: $4^2 = 16$ and $\sqrt{4} = 2$. Here $x^2$ is bigger.

They’re equal where both graphs pass through the same points, $(0, 0)$ and $(1, 1)$. So between $0$ and $1$, $\sqrt{x}$ is above $x^2$, and after $1$, $x^2$ is above.

Example 3: Behaviour near the asymptotes

Find $\dfrac{1}{x}$ for $x = 0.1$, $0.01$, and $-0.01$. Then describe what happens as $x$ gets very large.

Solution.

$$\frac{1}{0.1} = 10, \qquad \frac{1}{0.01} = 100, \qquad \frac{1}{-0.01} = -100$$

As $x$ gets close to $0$ from the right, the outputs shoot up; from the left, they shoot down. That’s the vertical asymptote at $x = 0$.

For large $x$, like $x = 1000$, $\dfrac{1}{x} = 0.001$: tiny, but never $0$. That’s the horizontal asymptote at $y = 0$.

Common mistakes

Drawing $y = \sqrt{x}$ on the left side. The domain is $x \ge 0$, so the graph starts at the origin and only goes right.

Letting $y = \dfrac{1}{x}$ touch an axis, or joining its two pieces. The axes are asymptotes, and $x = 0$ isn’t in the domain, so there’s a gap.

Giving $y = x^2$ negative outputs. Squaring never gives a negative number, so the range is $y \ge 0$.

Swapping the coordinates of key points. $\sqrt{4} = 2$, so the point is $(4, 2)$, not $(2, 4)$. The point $(2, 4)$ is on $y = x^2$.

Assuming $x^2$ is always bigger than $x$. For $x$ between $0$ and $1$, squaring makes a number smaller: $0.5^2 = 0.25$.

Practice

1. (Warm-up) Which parent function passes through $(4, 2)$?

Solution

$y = \sqrt{x}$, because $\sqrt{4} = 2$. (The others give $(4, 4)$, $(4, 16)$, and $\left(4, \tfrac{1}{4}\right)$.)

2. (Warm-up) Name the parent function that matches each table.

(a)

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$4$	$1$	$0$	$1$	$4$

(b)

$x$	$-2$	$-1$	$1$	$2$
$y$	$-0.5$	$-1$	$1$	$0.5$

Solution

(a) $y = x^2$. Each output is the input squared.

(b) $y = \dfrac{1}{x}$. Each output is $1$ divided by the input.

3. (Core) Which of the four parent functions are one-to-one? Explain.

Solution

$y = x$, $y = \sqrt{x}$, and $y = \dfrac{1}{x}$ are one-to-one: every output comes from only one input, so no horizontal line crosses their graphs more than once.

$y = x^2$ is many-to-one: for example, $x = 2$ and $x = -2$ both give $4$.

4. (Core) State the equations of the asymptotes of $y = \dfrac{1}{x}$, and explain why $x = 0$ isn’t in the domain.

Solution

Vertical asymptote $x = 0$ and horizontal asymptote $y = 0$.

$x = 0$ isn’t in the domain because $\dfrac{1}{0}$ would mean dividing by zero, which is undefined.

5. (Core) Find all values of $x$ where $x^2 = \sqrt{x}$.

Solution

Square both sides (both sides are $\ge 0$ for $x \ge 0$):

$$x^4 = x \quad\Rightarrow\quad x^4 - x = 0 \quad\Rightarrow\quad x(x^3 - 1) = 0$$

So $x = 0$ or $x^3 = 1$, which gives $x = 1$. Both check: $0^2 = \sqrt{0}$ and $1^2 = \sqrt{1}$.

The graphs meet at $(0, 0)$ and $(1, 1)$.

6. (Core) Without a calculator, put these in order from smallest to largest: $0.3$, $0.3^2$, $\sqrt{0.3}$.

Solution

For numbers between $0$ and $1$, squaring makes them smaller and square-rooting makes them bigger:

$$0.3^2 = 0.09 \;\lt\; 0.3 \;\lt\; \sqrt{0.3} \approx 0.548$$

7. (Challenge) For which values of $x$ is $\dfrac{1}{x} \gt x$? Use the graphs of $y = \dfrac{1}{x}$ and $y = x$ to help.

Solution

The graphs cross where $\dfrac{1}{x} = x$, so $x^2 = 1$ and $x = 1$ or $x = -1$. Together with the asymptote at $x = 0$, these split the number line into four parts. Test one value in each:

• $x = -2$: $\tfrac{1}{x} = -0.5$, which is greater than $-2$. ✓
• $x = -0.5$: $\tfrac{1}{x} = -2$, which is less than $-0.5$. ✗
• $x = 0.5$: $\tfrac{1}{x} = 2$, which is greater than $0.5$. ✓
• $x = 2$: $\tfrac{1}{x} = 0.5$, which is less than $2$. ✗

So $\dfrac{1}{x} \gt x$ when $x \lt -1$ or $0 \lt x \lt 1$.

8. (Challenge) The graph of $y = \sqrt{x}$ is the top half of a sideways parabola. Write the equation of that sideways parabola, and explain why the bottom half isn’t part of $y = \sqrt{x}$.

Solution

Squaring both sides of $y = \sqrt{x}$ gives $x = y^2$, which is a parabola opening to the right.

The bottom half of $x = y^2$ has negative $y$-values, like $(4, -2)$. But $\sqrt{x}$ always means the positive (principal) square root, so $\sqrt{4} = 2$, never $-2$. So only the top half, where $y \ge 0$, belongs to $y = \sqrt{x}$.