Reciprocal and Exponential Graphs

What are Reciprocal Graphs?

Have you ever noticed that if you share a fixed amount of money among more people, everyone gets a smaller share? This inverse relationship is the foundation of a reciprocal function.
A reciprocal function has the variable $x$ in the denominator, typically in the form $y = \frac{k}{x}$ (where $k$ is a constant and $x \neq 0$ ).
The graph of $y = \frac{1}{x}$ creates a specific "split-curve" shape known geometrically as a rectangular hyperbola.
It consists of two separate, identical curved parts called branches that never connect.

Sketching Reciprocal Graphs

Recognising the correct quadrants is the first step to an accurate sketch.
For $y = \frac{k}{x}$ where $k$ is positive, the branches sit in the first quadrant (top right) and third quadrant (bottom left).
If $k$ is negative (e.g., $y = -\frac{1}{x}$ ), the branches swap to the second (top left) and fourth (bottom right) quadrants.
The graph has rotational symmetry of order 2 (180°) about the origin $(0, 0)$ .
The basic graph has no $x$ -intercepts and no $y$ -intercepts, meaning it never touches or crosses either axis.
At $x = 0$ , there is a discontinuity because division by zero is mathematically undefined.
The graph is bounded by two asymptotes: straight lines the curve gets closer and closer to but never touches.
The horizontal asymptote is the $x$ -axis (the line $y = 0$ ), and the vertical asymptote is the $y$ -axis (the line $x = 0$ ).
If the graph is translated vertically to $y = \frac{1}{x} + c$ , the horizontal asymptote shifts to the line $y = c$ .

Solving with Reciprocal Graphs

Graphs are not just pictures; they are visual tools for solving complex algebraic equations.

Use the graph of $y = \frac{8}{x}$ to solve $\frac{8}{x} = 6$ .

Step 1: Recognise that finding where $\frac{8}{x} = 6$ is the same as finding where the curve $y = \frac{8}{x}$ intersects the line $y = 6$ .

Step 2: Draw a horizontal straight line at $y = 6$ on your graph.

Step 3: Locate the intersection point and read the corresponding $x$ -value.

$x \approx 1.33$

What are Exponential Graphs?

Unlike a piggy bank that grows by the same amount each week, a bacteria colony doubles in size over time, growing faster and faster.
An exponential function has the input variable $x$ occurring as an exponent (or index), usually in the form $y = k^x$ .
The graph of $y = k^x$ always passes through the $y$ -intercept at $(0, 1)$ , because any non-zero number to the power of $0$ is $1$ .
It also always passes through the point $(1, k)$ because $k^1 = k$ .
The $x$ -axis (the line $y = 0$ ) acts as a single horizontal asymptote for the curve.
If the equation takes the form $y = A \times k^x$ , the $y$ -intercept shifts to $(0, A)$ .

Growth vs Decay

A hot cup of coffee cooling down and an investment earning compound interest follow the exact same mathematical rules.
Exponential growth occurs when the multiplier (or growth factor) $k > 1$ .
The growth graph is a smooth curve starting near the negative $x$ -axis and increasing with an ever-steepening gradient as it moves right.
Exponential decay occurs when $0 < k < 1$ .
The decay graph is a smooth curve that falls from left to right, approaching the positive $x$ -axis, acting as a reflection of the growth graph in the $y$ -axis.
For both growth and decay, the graph stays entirely in the first and second quadrants because the range is always $y > 0$ .

Finding the Multiplier ( $k$ )

Sometimes you only have a single snapshot of data to figure out an entire mathematical rule.

An exponential graph $y = k^x$ passes through the point $(2, 16)$ . Find the value of $k$ .

Step 1: Substitute the given $x$ and $y$ coordinates into the equation.

$16 = k^2$

Step 2: Solve for $k$ .

$k = \sqrt{16}$

Step 3: State the final positive value (since $k > 0$ for standard Edexcel exponential functions).

$k = 4$

An exponential decay graph $y = k^x$ passes through $(-2, 16)$ . Find $k$ .

Step 1: Substitute the given coordinates into the equation.

$16 = k^{-2}$

Step 2: Use index laws to rewrite the negative exponent.

$16 = \frac{1}{k^2}$

Step 3: Rearrange the equation to make $k^2$ the subject.

$k^2 = \frac{1}{16}$

Step 4: Square root to find the positive base value $k$ .

$k = 0.25$

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

Exam Tips

Common Mistake
Students often draw reciprocal graphs that accidentally touch or cross the axes, or "flick" away at the ends; this loses marks for not showing asymptotic behaviour clearly.
2
In exam sketches, never join points with straight line segments; Edexcel strictly requires smooth curves for both reciprocal and exponential graphs.
3
If asked to interpret the $y$ -intercept of an exponential graph in a real-world context, state that it represents the initial amount (e.g., starting population) at time $t = 0$ .
4
Reciprocal graphs are sometimes confused with quadratic or cubic graphs if you only look at one quadrant, so always check for the "split" branches across two quadrants.
5
In Paper 2 and Paper 3, use the 'TABLE' mode on your scientific calculator to quickly generate accurate coordinate points for plotting.

Key Terms(12)

Reciprocal function: A function where the variable $x$ is in the denominator, typically in the form $y = \frac{k}{x}$ where $k$ is a constant and $x \neq 0$ .
Rectangular hyperbola: The geometric name for the specific split-curve shape produced by plotting reciprocal functions.
Branches: The two separate, identical curved parts that make up a reciprocal graph.
Quadrants: The four regions divided by the $x$ and $y$ axes on a coordinate grid.
Rotational symmetry of order 2: A shape that looks exactly the same twice during a full 360-degree rotation, specifically at 180 degrees.
Discontinuity: A point where a mathematical function is not defined, such as at $x = 0$ for a reciprocal graph due to division by zero.
Asymptotes: Straight lines that a curve approaches closer and closer as it tends toward infinity, but never actually meets or crosses.
Exponential function: A function where the input variable $x$ occurs as an exponent (or index), usually in the form $y = k^x$ .
y-intercept: The point where a graph crosses the vertical $y$ -axis, occurring when $x = 0$ .
Exponential growth: A relationship where a quantity increases at an ever-steepening rate, occurring when the multiplier is greater than $1$ .
Exponential decay: A relationship where a quantity decreases by a constant percentage or scale factor over equal increments of time, occurring when the multiplier is between $0$ and $1$ .
Multiplier: The constant base value ( $k$ ) in an exponential function that determines the rate of growth or decay.

Previous NoteLinear, Quadratic and Cubic Graphs Next NoteTrigonometric Graphs

Back to Function Types

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Key Terms(12)

Reciprocal function: A function where the variable $x$ is in the denominator, typically in the form $y = \frac{k}{x}$ where $k$ is a constant and $x \neq 0$ .
Rectangular hyperbola: The geometric name for the specific split-curve shape produced by plotting reciprocal functions.
Branches: The two separate, identical curved parts that make up a reciprocal graph.
Quadrants: The four regions divided by the $x$ and $y$ axes on a coordinate grid.
Rotational symmetry of order 2: A shape that looks exactly the same twice during a full 360-degree rotation, specifically at 180 degrees.
Discontinuity: A point where a mathematical function is not defined, such as at $x = 0$ for a reciprocal graph due to division by zero.
Asymptotes: Straight lines that a curve approaches closer and closer as it tends toward infinity, but never actually meets or crosses.
Exponential function: A function where the input variable $x$ occurs as an exponent (or index), usually in the form $y = k^x$ .
y-intercept: The point where a graph crosses the vertical $y$ -axis, occurring when $x = 0$ .
Exponential growth: A relationship where a quantity increases at an ever-steepening rate, occurring when the multiplier is greater than $1$ .
Exponential decay: A relationship where a quantity decreases by a constant percentage or scale factor over equal increments of time, occurring when the multiplier is between $0$ and $1$ .
Multiplier: The constant base value ( $k$ ) in an exponential function that determines the rate of growth or decay.

Reciprocal and Exponential Graphs

What are Reciprocal Graphs?

Have you ever noticed that if you share a fixed amount of money among more people, everyone gets a smaller share? This inverse relationship is the foundation of a reciprocal function.
A reciprocal function has the variable $x$ in the denominator, typically in the form $y = \frac{k}{x}$ (where $k$ is a constant and $x \neq 0$ ).
The graph of $y = \frac{1}{x}$ creates a specific "split-curve" shape known geometrically as a rectangular hyperbola.
It consists of two separate, identical curved parts called branches that never connect.

Sketching Reciprocal Graphs

Recognising the correct quadrants is the first step to an accurate sketch.
For $y = \frac{k}{x}$ where $k$ is positive, the branches sit in the first quadrant (top right) and third quadrant (bottom left).
If $k$ is negative (e.g., $y = -\frac{1}{x}$ ), the branches swap to the second (top left) and fourth (bottom right) quadrants.
The graph has rotational symmetry of order 2 (180°) about the origin $(0, 0)$ .
The basic graph has no $x$ -intercepts and no $y$ -intercepts, meaning it never touches or crosses either axis.
At $x = 0$ , there is a discontinuity because division by zero is mathematically undefined.
The graph is bounded by two asymptotes: straight lines the curve gets closer and closer to but never touches.
The horizontal asymptote is the $x$ -axis (the line $y = 0$ ), and the vertical asymptote is the $y$ -axis (the line $x = 0$ ).
If the graph is translated vertically to $y = \frac{1}{x} + c$ , the horizontal asymptote shifts to the line $y = c$ .

Solving with Reciprocal Graphs

Graphs are not just pictures; they are visual tools for solving complex algebraic equations.

Use the graph of $y = \frac{8}{x}$ to solve $\frac{8}{x} = 6$ .

Step 1: Recognise that finding where $\frac{8}{x} = 6$ is the same as finding where the curve $y = \frac{8}{x}$ intersects the line $y = 6$ .

Step 2: Draw a horizontal straight line at $y = 6$ on your graph.

Step 3: Locate the intersection point and read the corresponding $x$ -value.

$x \approx 1.33$

What are Exponential Graphs?

Unlike a piggy bank that grows by the same amount each week, a bacteria colony doubles in size over time, growing faster and faster.
An exponential function has the input variable $x$ occurring as an exponent (or index), usually in the form $y = k^x$ .
The graph of $y = k^x$ always passes through the $y$ -intercept at $(0, 1)$ , because any non-zero number to the power of $0$ is $1$ .
It also always passes through the point $(1, k)$ because $k^1 = k$ .
The $x$ -axis (the line $y = 0$ ) acts as a single horizontal asymptote for the curve.
If the equation takes the form $y = A \times k^x$ , the $y$ -intercept shifts to $(0, A)$ .

Growth vs Decay

A hot cup of coffee cooling down and an investment earning compound interest follow the exact same mathematical rules.
Exponential growth occurs when the multiplier (or growth factor) $k > 1$ .
The growth graph is a smooth curve starting near the negative $x$ -axis and increasing with an ever-steepening gradient as it moves right.
Exponential decay occurs when $0 < k < 1$ .
The decay graph is a smooth curve that falls from left to right, approaching the positive $x$ -axis, acting as a reflection of the growth graph in the $y$ -axis.
For both growth and decay, the graph stays entirely in the first and second quadrants because the range is always $y > 0$ .

Finding the Multiplier ( $k$ )

Sometimes you only have a single snapshot of data to figure out an entire mathematical rule.

An exponential graph $y = k^x$ passes through the point $(2, 16)$ . Find the value of $k$ .

Step 1: Substitute the given $x$ and $y$ coordinates into the equation.

$16 = k^2$

Step 2: Solve for $k$ .

$k = \sqrt{16}$

Step 3: State the final positive value (since $k > 0$ for standard Edexcel exponential functions).

$k = 4$

An exponential decay graph $y = k^x$ passes through $(-2, 16)$ . Find $k$ .

Step 1: Substitute the given coordinates into the equation.

$16 = k^{-2}$

Step 2: Use index laws to rewrite the negative exponent.

$16 = \frac{1}{k^2}$

Step 3: Rearrange the equation to make $k^2$ the subject.

$k^2 = \frac{1}{16}$

Step 4: Square root to find the positive base value $k$ .

$k = 0.25$

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 draw reciprocal graphs that accidentally touch or cross the axes, or "flick" away at the ends; this loses marks for not showing asymptotic behaviour clearly.
2
In exam sketches, never join points with straight line segments; Edexcel strictly requires smooth curves for both reciprocal and exponential graphs.
3
If asked to interpret the $y$ -intercept of an exponential graph in a real-world context, state that it represents the initial amount (e.g., starting population) at time $t = 0$ .
4
Reciprocal graphs are sometimes confused with quadratic or cubic graphs if you only look at one quadrant, so always check for the "split" branches across two quadrants.
5
In Paper 2 and Paper 3, use the 'TABLE' mode on your scientific calculator to quickly generate accurate coordinate points for plotting.

Key Terms(12)

Reciprocal function: A function where the variable $x$ is in the denominator, typically in the form $y = \frac{k}{x}$ where $k$ is a constant and $x \neq 0$ .
Rectangular hyperbola: The geometric name for the specific split-curve shape produced by plotting reciprocal functions.
Branches: The two separate, identical curved parts that make up a reciprocal graph.
Quadrants: The four regions divided by the $x$ and $y$ axes on a coordinate grid.
Rotational symmetry of order 2: A shape that looks exactly the same twice during a full 360-degree rotation, specifically at 180 degrees.
Discontinuity: A point where a mathematical function is not defined, such as at $x = 0$ for a reciprocal graph due to division by zero.
Asymptotes: Straight lines that a curve approaches closer and closer as it tends toward infinity, but never actually meets or crosses.
Exponential function: A function where the input variable $x$ occurs as an exponent (or index), usually in the form $y = k^x$ .
y-intercept: The point where a graph crosses the vertical $y$ -axis, occurring when $x = 0$ .
Exponential growth: A relationship where a quantity increases at an ever-steepening rate, occurring when the multiplier is greater than $1$ .
Exponential decay: A relationship where a quantity decreases by a constant percentage or scale factor over equal increments of time, occurring when the multiplier is between $0$ and $1$ .
Multiplier: The constant base value ( $k$ ) in an exponential function that determines the rate of growth or decay.

Previous NoteLinear, Quadratic and Cubic Graphs Next NoteTrigonometric Graphs

Back to Function Types

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

Key Terms(12)

Reciprocal function: A function where the variable $x$ is in the denominator, typically in the form $y = \frac{k}{x}$ where $k$ is a constant and $x \neq 0$ .
Rectangular hyperbola: The geometric name for the specific split-curve shape produced by plotting reciprocal functions.
Branches: The two separate, identical curved parts that make up a reciprocal graph.
Quadrants: The four regions divided by the $x$ and $y$ axes on a coordinate grid.
Rotational symmetry of order 2: A shape that looks exactly the same twice during a full 360-degree rotation, specifically at 180 degrees.
Discontinuity: A point where a mathematical function is not defined, such as at $x = 0$ for a reciprocal graph due to division by zero.
Asymptotes: Straight lines that a curve approaches closer and closer as it tends toward infinity, but never actually meets or crosses.
Exponential function: A function where the input variable $x$ occurs as an exponent (or index), usually in the form $y = k^x$ .
y-intercept: The point where a graph crosses the vertical $y$ -axis, occurring when $x = 0$ .
Exponential growth: A relationship where a quantity increases at an ever-steepening rate, occurring when the multiplier is greater than $1$ .
Exponential decay: A relationship where a quantity decreases by a constant percentage or scale factor over equal increments of time, occurring when the multiplier is between $0$ and $1$ .
Multiplier: The constant base value ( $k$ ) in an exponential function that determines the rate of growth or decay.

Reciprocal and Exponential Graphs

What are Reciprocal Graphs?

Sketching Reciprocal Graphs

Solving with Reciprocal Graphs

What are Exponential Graphs?

Growth vs Decay

Finding the Multiplier (kkk)

Sign up to continue reading

Exam Tips

Key Terms(12)

Key Terms(12)

Reciprocal and Exponential Graphs

What are Reciprocal Graphs?

Sketching Reciprocal Graphs

Solving with Reciprocal Graphs

What are Exponential Graphs?

Growth vs Decay

Finding the Multiplier (kkk)

Sign up to continue reading

Exam Tips

Key Terms(12)

Key Terms(12)

Finding the Multiplier ( $k$ )

Finding the Multiplier ( $k$ )