Cubic Functions of a Particular Form Revision Notes for VCE SSCE Mathematical Methods

Cubic Functions of a Particular Form

Introduction

This note focuses on cubic functions that can be written in the form:

f(x) = a(x - h)^3 + k

Unlike quadratic functions, which all have the same basic parabola shape, not all cubic functions look alike. However, cubic functions of the form $f(x) = a(x - h)^3 + k$ share a common characteristic: they can all be created by applying transformations to the basic cubic graph $f(x) = x^3$ .

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Understanding transformations is key to working with cubic functions. Each parameter in the form $f(x) = a(x - h)^3 + k$ controls a specific transformation:

$a$ controls dilation and reflection
$h$ controls horizontal translation
$k$ controls vertical translation

For example, the graph of $f(x) = (x - 1)^3 + 3$ is obtained from the graph of $f(x) = x^3$ by translating it 1 unit to the right and 3 units up.

Transformations of the graph of $f(x) = x^3$

Dilations and reflections

The value of $a$ in the function $f(x) = a(x - h)^3 + k$ affects how narrow or broad the graph appears, without changing its fundamental S-shape.

When a > 1, the graph becomes steeper (narrower).

When 0 < a < 1, the graph becomes less steep (broader).

When a < 0, the graph is reflected in an axis.

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An Interesting Property of Cubic Functions

Reflecting in the $x$ -axis produces the same result as reflecting in the $y$ -axis. This happens because:

(-x)^3 = -x^3

This unique property makes cubic functions behave differently from other polynomial functions.

The following graphs illustrate these dilations:

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Domain and Range of Cubic Functions

The implied domain of all cubic functions is $\mathbb{R}$ (all real numbers) and the range is also $\mathbb{R}$ .

This means cubic functions can accept any real number as input and can produce any real number as output.

Point of inflection

The most important feature of these cubic graphs is the point of inflection. This is the point where the gradient (slope) is zero - you can think of it as the "flat point" of the graph.

For the basic cubic function $y = x^3$ , the point of inflection is at the origin $(0, 0)$ .

Vertical translations

When you add or subtract a constant $k$ to $y = x^3$ , the graph moves up or down vertically.

The graph of $y = x^3 + k$ is the basic cubic graph moved k units upward (when $k > 0$ ) or downward (when $k < 0$ ).

After a vertical translation, the point of inflection becomes $(0, k)$ .

This represents a translation of $k$ units in the positive $y$ -axis direction.

Horizontal translations

The graph of $y = (x - h)^3$ is the basic cubic graph moved h units to the right (when $h > 0$ ) or to the left (when $h < 0$ ).

After a horizontal translation, the point of inflection moves to $(h, 0)$ .

This represents a translation of $h$ units in the positive $x$ -axis direction.

General form and sketching technique

For the graph of a cubic function of the form:

y = a(x - h)^3 + k

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Point of Inflection Formula

The point of inflection is at $(h, k)$ .

The values in the brackets and the added constant directly give you the coordinates of the point of inflection!

Sketching technique: When sketching cubic graphs of this form:

First identify the point of inflection $(h, k)$
Find the $x$ -axis intercept by setting $y = 0$
Find the $y$ -axis intercept by setting $x = 0$
Draw the smooth S-shaped curve passing through these points

Worked example: Sketching a cubic function

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Worked Example: Sketching a Cubic Function

Sketch the graph of the function $y = (x - 2)^3 + 4$ .

Solution:

The graph of $y = x^3$ is translated 2 units to the right and 4 units up.

Point of inflection is (2, 4).

Finding the $x$ -axis intercept:

Let $y = 0$

0 = (x - 2)^3 + 4

-4 = (x - 2)^3

\sqrt[3]{-4} = x - 2

x = 2 + \sqrt[3]{-4}

x \approx 0.413

Finding the $y$ -axis intercept:

Let $x = 0$

y = (0 - 2)^3 + 4

y = -8 + 4

y = -4

The graph passes through the points $(2 + \sqrt[3]{-4}, 0)$ , $(2, 4)$ , and $(0, -4)$ .

The cube root function $f(x) = x^{\frac{1}{3}}$

Functions with rules of the form $f(x) = a(x - h)^3 + k$ are one-to-one functions. This means each of these functions has an inverse function.

The inverse function of $f(x) = x^3$ is:

f^{-1}(x) = x^{\frac{1}{3}}

This can also be written as $f^{-1}(x) = \sqrt[3]{x}$ (the cube root of $x$ ).

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Inverse Functions Relationship

The cube root function "undoes" the cubing operation. If you cube a number and then take the cube root, you get back to the original number:

\sqrt[3]{x^3} = x

Similarly, if you take the cube root and then cube the result:

(\sqrt[3]{x})^3 = x

The following graph shows both functions:

The graph of $y = x^{\frac{1}{3}}$ is instantaneously vertical at $x = 0$ . The graphs of $y = x^3$ and $y = x^{\frac{1}{3}}$ intersect at $(1, 1)$ and $(-1, -1)$ .

Worked example: Sketching a cube root function

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Worked Example: Sketching a Cube Root Function

Sketch the graph of $y = (x - 1)^{\frac{1}{3}} - 2$ .

Solution:

Finding the $y$ -axis intercept:

When $x = 0$ :

y = (-1)^{\frac{1}{3}} - 2

y = -1 - 2

y = -3

Finding the $x$ -axis intercept:

When $y = 0$ :

(x - 1)^{\frac{1}{3}} - 2 = 0

(x - 1)^{\frac{1}{3}} = 2

x - 1 = 2^3

x = 9

Understanding the transformation:

The graph of $y = (x - 1)^{\frac{1}{3}} - 2$ is the graph of $y = x^{\frac{1}{3}}$ translated 1 unit to the right and 2 units down.

The graph passes through $(0, -3)$ and $(9, 0)$ .

Worked example: Finding inverse functions

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Worked Example: Finding the Inverse Function

Find the inverse function $f^{-1}$ of $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = 2(x - 4)^3 + 3$ .

Solution:

Step 1: Interchange $x$ and $y$

Remember that $(x, y) \in f$ if and only if $(y, x) \in f^{-1}$ .

Starting with: $y = 2(x - 4)^3 + 3$

Interchange: $x = 2(y - 4)^3 + 3$

Step 2: Solve for $y$

x - 3 = 2(y - 4)^3

\frac{x - 3}{2} = (y - 4)^3

The opposite operation to cubing is taking the cube root:

y - 4 = \left(\frac{x - 3}{2}\right)^{\frac{1}{3}}

y = \left(\frac{x - 3}{2}\right)^{\frac{1}{3}} + 4

Answer:

Therefore: $f^{-1} : \mathbb{R} \to \mathbb{R}$ , $f^{-1}(x) = \left(\frac{x - 3}{2}\right)^{\frac{1}{3}} + 4$

Remember!

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Key Points to Remember:

Cubic functions of the form $y = a(x - h)^3 + k$ have their point of inflection at $(h, k)$ .
The graph of $y = a(x - h)^3 + k$ is the graph of $y = ax^3$ translated $h$ units horizontally and $k$ units vertically.
To sketch these cubic graphs: identify the point of inflection first, then find the $x$ -axis and $y$ -axis intercepts.
The domain and range of all cubic functions is $\mathbb{R}$ (all real numbers).
The inverse function of $f(x) = x^3$ is $f^{-1}(x) = x^{\frac{1}{3}}$ (the cube root function). These are inverse operations that "undo" each other.

Cubic Functions of a Particular Form (VCE SSCE Mathematical Methods): Revision Notes