Graphs of Cubic and Reciprocal Functions

Sketching Cubic Functions

Think of the track of a rollercoaster that climbs steeply, levels out for a split second, and then continues climbing. This "wiggle" is the signature shape of a simple cubic function, such as $y = x^3$ .
A cubic function is a polynomial where the highest power of $x$ is 3. The graph forms a smooth, continuous "S-shaped" curve.
For a positive cubic ( $y = x^3$ ), the curve starts in the third quadrant (bottom-left) and rises into the first quadrant (top-right). At the origin $(0, 0)$ , the gradient momentarily flattens out at a point of inflection.
A negative cubic ( $y = -x^3$ ) is a reflection of $y = x^3$ and forms a strictly "downhill" curve. It starts in the second quadrant (top-left) and ends in the fourth quadrant (bottom-right).

Finding the Intercepts for Cubic Graphs

When sketching $y = x^3 + k$ , the curve is translated vertically up or down by $k$ units.
You must always label the $y$ -intercept, which is the point where the graph crosses the $y$ -axis.
You should also calculate and mark any roots, which are the $x$ -intercepts where the graph crosses the $x$ -axis.

Worked Example: Sketching a Cubic

Sketch the graph of $y = x^3 - 8$ .

Step 1: Find the $y$ -intercept by setting $x = 0$ .

$y = 0^3 - 8 = -8$
Label this coordinate: $(0, -8)$

Step 2: Find the root by setting $y = 0$ .

$0 = x^3 - 8$
$x^3 = 8 \implies x = 2$
Label this coordinate: $(2, 0)$

Step 3: Describe the sketch.

Draw a smooth "uphill" S-shape passing through the $y$ -axis at $(0, -8)$ and the $x$ -axis at $(2, 0)$ , ensuring the point of inflection is at $(0, -8)$ .

Sketching Reciprocal Functions

If you share a pizza, the more friends you invite, the smaller each slice gets — but the slice size never quite reaches zero. This inverse relationship is visualised by a reciprocal function like $y = \frac{1}{x}$ .
The graph of a reciprocal function forms a hyperbola, which consists of two separate, non-connecting branches.
For positive reciprocal functions (like $y = \frac{1}{x}$ ), these branches sit in the first and third quadrants. For negative reciprocals (like $y = -\frac{1}{x}$ ), they appear in the second and fourth quadrants.

Asymptotes and Branches

The defining feature of a reciprocal graph is its asymptotes. An asymptote is a straight line that a curve approaches increasingly closely but never actually touches.
For $y = \frac{k}{x}$ , the vertical asymptote is the $y$ -axis ( $x = 0$ ) and the horizontal asymptote is the $x$ -axis ( $y = 0$ ).
Larger values of $k$ stretch the branches further away from the origin. You must draw both freehand branches smoothly, ensuring they "hug" the axes without ever making contact.

Worked Example: Sketching a Reciprocal Function

Sketch the graph of $y = \frac{2}{x}$ .

Step 1: Identify the asymptotes.

The vertical asymptote is the $y$ -axis ( $x = 0$ ).
The horizontal asymptote is the $x$ -axis ( $y = 0$ ).

Step 2: Find key coordinate points to guide the shape.

When $x = 1$ , $y = \frac{2}{1} = 2$ . Point: $(1, 2)$
When $x = -1$ , $y = \frac{2}{-1} = -2$ . Point: $(-1, -2)$

Step 3: Describe the sketch.

Draw two smooth, separate branches in the first quadrant and third quadrant passing through the key points $(1, 2)$ and $(-1, -2)$ .
Ensure the branches "hug" the axes without touching or crossing them.

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Exam Tips

Common Mistake
Students often confuse the graph of $y = x^3$ with a quadratic $y = x^2$ — ensure you draw the 'tail' extending into the third quadrant rather than a U-shape.
2
AQA examiners require a single, continuous smooth curve for cubics; you will lose marks for jagged lines or drawing ends that 'bend back' on themselves.
3
The 'Asymptote Rule' means you will lose marks if your reciprocal curve touches or crosses the $x$ or $y$ axis, or if it flicks away from the axes at the ends.
4
When asked to 'sketch' a graph, always label the $y$ -intercept and roots with their full coordinates (e.g., $(0, -4)$ rather than just $-4$ on the axis).

Key Terms(8)

Cubic function: A polynomial function where the highest power of the variable $x$ is 3 (e.g., $y = x^3$ ).
Quadrant: One of the four quarters of the coordinate plane, separated by the $x$ and $y$ axes.
Point of inflection: The point on a curve where the curvature changes direction, such as the origin in $y = x^3$ where the gradient momentarily levels off.
y-intercept: The point where the graph crosses the $y$ -axis, found by setting $x = 0$ .
Root: The point where the graph crosses the $x$ -axis, found by setting $y = 0$ .
Reciprocal function: A function of the form $y = \frac{k}{x}$ , where $k$ is a non-zero constant and $x \neq 0$ .
Hyperbola: The specific geometric shape formed by the two discontinuous branches of a reciprocal function.
Asymptote: A straight line that a curve approaches increasingly closely but never actually touches or crosses.

Previous NoteGraphs of Linear and Quadratic Functions Next NoteGraphs of Exponential Functions

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Key Terms(8)

Cubic function: A polynomial function where the highest power of the variable $x$ is 3 (e.g., $y = x^3$ ).
Quadrant: One of the four quarters of the coordinate plane, separated by the $x$ and $y$ axes.
Point of inflection: The point on a curve where the curvature changes direction, such as the origin in $y = x^3$ where the gradient momentarily levels off.
y-intercept: The point where the graph crosses the $y$ -axis, found by setting $x = 0$ .
Root: The point where the graph crosses the $x$ -axis, found by setting $y = 0$ .
Reciprocal function: A function of the form $y = \frac{k}{x}$ , where $k$ is a non-zero constant and $x \neq 0$ .
Hyperbola: The specific geometric shape formed by the two discontinuous branches of a reciprocal function.
Asymptote: A straight line that a curve approaches increasingly closely but never actually touches or crosses.

Graphs of Cubic and Reciprocal Functions

Sketching Cubic Functions

Think of the track of a rollercoaster that climbs steeply, levels out for a split second, and then continues climbing. This "wiggle" is the signature shape of a simple cubic function, such as $y = x^3$ .
A cubic function is a polynomial where the highest power of $x$ is 3. The graph forms a smooth, continuous "S-shaped" curve.
For a positive cubic ( $y = x^3$ ), the curve starts in the third quadrant (bottom-left) and rises into the first quadrant (top-right). At the origin $(0, 0)$ , the gradient momentarily flattens out at a point of inflection.
A negative cubic ( $y = -x^3$ ) is a reflection of $y = x^3$ and forms a strictly "downhill" curve. It starts in the second quadrant (top-left) and ends in the fourth quadrant (bottom-right).

Finding the Intercepts for Cubic Graphs

When sketching $y = x^3 + k$ , the curve is translated vertically up or down by $k$ units.
You must always label the $y$ -intercept, which is the point where the graph crosses the $y$ -axis.
You should also calculate and mark any roots, which are the $x$ -intercepts where the graph crosses the $x$ -axis.

Worked Example: Sketching a Cubic

Sketch the graph of $y = x^3 - 8$ .

Step 1: Find the $y$ -intercept by setting $x = 0$ .

$y = 0^3 - 8 = -8$
Label this coordinate: $(0, -8)$

Step 2: Find the root by setting $y = 0$ .

$0 = x^3 - 8$
$x^3 = 8 \implies x = 2$
Label this coordinate: $(2, 0)$

Step 3: Describe the sketch.

Draw a smooth "uphill" S-shape passing through the $y$ -axis at $(0, -8)$ and the $x$ -axis at $(2, 0)$ , ensuring the point of inflection is at $(0, -8)$ .

Sketching Reciprocal Functions

If you share a pizza, the more friends you invite, the smaller each slice gets — but the slice size never quite reaches zero. This inverse relationship is visualised by a reciprocal function like $y = \frac{1}{x}$ .
The graph of a reciprocal function forms a hyperbola, which consists of two separate, non-connecting branches.
For positive reciprocal functions (like $y = \frac{1}{x}$ ), these branches sit in the first and third quadrants. For negative reciprocals (like $y = -\frac{1}{x}$ ), they appear in the second and fourth quadrants.

Asymptotes and Branches

The defining feature of a reciprocal graph is its asymptotes. An asymptote is a straight line that a curve approaches increasingly closely but never actually touches.
For $y = \frac{k}{x}$ , the vertical asymptote is the $y$ -axis ( $x = 0$ ) and the horizontal asymptote is the $x$ -axis ( $y = 0$ ).
Larger values of $k$ stretch the branches further away from the origin. You must draw both freehand branches smoothly, ensuring they "hug" the axes without ever making contact.

Worked Example: Sketching a Reciprocal Function

Sketch the graph of $y = \frac{2}{x}$ .

Step 1: Identify the asymptotes.

The vertical asymptote is the $y$ -axis ( $x = 0$ ).
The horizontal asymptote is the $x$ -axis ( $y = 0$ ).

Step 2: Find key coordinate points to guide the shape.

When $x = 1$ , $y = \frac{2}{1} = 2$ . Point: $(1, 2)$
When $x = -1$ , $y = \frac{2}{-1} = -2$ . Point: $(-1, -2)$

Step 3: Describe the sketch.

Draw two smooth, separate branches in the first quadrant and third quadrant passing through the key points $(1, 2)$ and $(-1, -2)$ .
Ensure the branches "hug" the axes without touching or crossing them.

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often confuse the graph of $y = x^3$ with a quadratic $y = x^2$ — ensure you draw the 'tail' extending into the third quadrant rather than a U-shape.
2
AQA examiners require a single, continuous smooth curve for cubics; you will lose marks for jagged lines or drawing ends that 'bend back' on themselves.
3
The 'Asymptote Rule' means you will lose marks if your reciprocal curve touches or crosses the $x$ or $y$ axis, or if it flicks away from the axes at the ends.
4
When asked to 'sketch' a graph, always label the $y$ -intercept and roots with their full coordinates (e.g., $(0, -4)$ rather than just $-4$ on the axis).

Key Terms(8)

Cubic function: A polynomial function where the highest power of the variable $x$ is 3 (e.g., $y = x^3$ ).
Quadrant: One of the four quarters of the coordinate plane, separated by the $x$ and $y$ axes.
Point of inflection: The point on a curve where the curvature changes direction, such as the origin in $y = x^3$ where the gradient momentarily levels off.
y-intercept: The point where the graph crosses the $y$ -axis, found by setting $x = 0$ .
Root: The point where the graph crosses the $x$ -axis, found by setting $y = 0$ .
Reciprocal function: A function of the form $y = \frac{k}{x}$ , where $k$ is a non-zero constant and $x \neq 0$ .
Hyperbola: The specific geometric shape formed by the two discontinuous branches of a reciprocal function.
Asymptote: A straight line that a curve approaches increasingly closely but never actually touches or crosses.

Previous NoteGraphs of Linear and Quadratic Functions Next NoteGraphs of Exponential Functions

Back to Function Graphs

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(8)

Cubic function: A polynomial function where the highest power of the variable $x$ is 3 (e.g., $y = x^3$ ).
Quadrant: One of the four quarters of the coordinate plane, separated by the $x$ and $y$ axes.
Point of inflection: The point on a curve where the curvature changes direction, such as the origin in $y = x^3$ where the gradient momentarily levels off.
y-intercept: The point where the graph crosses the $y$ -axis, found by setting $x = 0$ .
Root: The point where the graph crosses the $x$ -axis, found by setting $y = 0$ .
Reciprocal function: A function of the form $y = \frac{k}{x}$ , where $k$ is a non-zero constant and $x \neq 0$ .
Hyperbola: The specific geometric shape formed by the two discontinuous branches of a reciprocal function.
Asymptote: A straight line that a curve approaches increasingly closely but never actually touches or crosses.