The diagrams $A$, $B$, $C$, $D$, $E$ and $F$ show six graphs of different functions. Select the diagram that best shows that

1. Express the inverse proportion relationship as a linear equation

If $y$ is inversely proportional to $x$, then $y = \frac{k}{x}$ for some constant $k > 0$.
This can be rewritten in linear form as $y = k \left(\frac{1}{x}\right)$.

2. Identify the corresponding graph

Plotting $y$ against $\frac{1}{x}$ yields a straight line passing through the origin with gradient $k$. Diagram A shows a straight line graph of $y$ against $\frac{1}{x}$ starting from the origin.

1. Identify the key features of the graph of $y = x^n$ for an odd integer $n > 1$

For $y = x^n$ where $n$ is an odd integer greater than 1 (e.g., $n=3$):

• The graph passes through the origin $(0,0)$.
• It is an odd function, meaning it has rotational symmetry about the origin (it lies in the first and third quadrants).
• The gradient $\frac{dy}{dx} = nx^{n-1}$ is $0$ at $x=0$, meaning the curve has a horizontal tangent at the origin.
• As $x$ increases, the gradient increases, making the curve steeper.

2. Select the matching diagram

Diagram E shows a curve passing through the origin, lying in the first and third quadrants, and flattening out (horizontal tangent) at the origin before becoming steeper. This matches the properties of $y = x^n$ for odd $n > 1$. (Diagram F has a vertical tangent at the origin, which corresponds to $y = x^{1/n}$).

1. Identify the characteristics of the graph of $y = n^x$ for $n > 1$

The function $y = n^x$ where $n > 1$ represents exponential growth. The graph must lie entirely above the $x$-axis ($y > 0$), pass through $(0, 1)$, and increase rapidly as $x$ increases.

2. Match the characteristics to the given diagrams

Diagram B shows a curve that is entirely above the $x$-axis, has a positive $y$-intercept, and increases exponentially from left to right. This matches the properties of $y = n^x$.