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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 is a constant and ).

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 .

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 at , because any non-zero number to the power of is .

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 -axis, acting as a reflection of the growth graph in the -axis.

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 $k$ .

$k = 0.25$

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.

In exam sketches, never join points with straight line segments; Edexcel strictly requires smooth curves for both reciprocal and exponential graphs.

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 $t = 0$ .

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.

In Paper 2 and Paper 3, use the 'TABLE' mode on your scientific calculator to quickly generate accurate coordinate points for plotting.

A function where the variable $x$ is in the denominator, typically in the form $y = \frac{k}{x}$ where $k$ is a constant and $x \neq 0$ .

The geometric name for the specific split-curve shape produced by plotting reciprocal functions.

The two separate, identical curved parts that make up a reciprocal graph.

The four regions divided by the $x$ and $y$ axes on a coordinate grid.

A shape that looks exactly the same twice during a full 360-degree rotation, specifically at 180 degrees.

A point where a mathematical function is not defined, such as at $x = 0$ for a reciprocal graph due to division by zero.

Straight lines that a curve approaches closer and closer as it tends toward infinity, but never actually meets or crosses.

A function where the input variable $x$ occurs as an exponent (or index), usually in the form $y = k^x$ .

The point where a graph crosses the vertical $y$ -axis, occurring when $x = 0$ .

A relationship where a quantity increases at an ever-steepening rate, occurring when the multiplier is greater than $1$ .

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 $1$ .

The constant base value ( $k$ ) in an exponential function that determines the rate of growth or decay.