Reciprocal and Exponential Graphs

The reciprocal function

The reciprocal of a non-zero number $x$ is $1/x$. The function

$y=\frac{1}{x}$

therefore pairs each permitted value of $x$ with its reciprocal. A table gives useful guide points:

Plotting these coordinates produces two separate curved branches.

When $x$ is positive, $1/x$ is positive, so one branch lies in the first quadrant. When $x$ is negative, $1/x$ is negative, so the other lies in the third quadrant. Corresponding points such as $(2,\frac12)$ and show that the graph has rotational symmetry about the origin.

Why the reciprocal graph avoids both axes

The restriction $x\ne0$ is essential. Division by zero is undefined, so the graph has no point with $x=0$ and therefore no y-intercept. As positive $x$ gets closer to zero, becomes increasingly large; from the negative side, it becomes increasingly negative. The line is a vertical .

The output can never equal zero either: no finite non-zero value of $x$ makes $1/x=0$. As the magnitude of $x$ grows, the magnitude of $1/x$ becomes smaller and the branches approach the x-axis. Thus $y=0$ is a horizontal asymptote, not part of the graph.

To sketch the function, mark a few points on each side of zero, draw two smooth branches and show each branch approaching both axes without touching them.

Exponential functions

In an exponential function

$y=k^x,$

$x$ is the exponent and the base $k$ is a positive constant. Integer values of make a convenient table of guide points. Two index facts are particularly important:

$k^0=1 \qquad\text{and}\qquad k^{-n}=\frac{1}{k^n}.$

Consequently, every graph $y=k^x$ with $k>0$ passes through $(0,1)$. Its values are always positive, so the curve remains above the x-axis.

If $k>1$, multiplying by $k$ each time $x$ increases by 1 makes the graph rise from left to right. For example, the outputs of $y=2^x$ double at each integer step. As becomes increasingly negative, the outputs approach zero.

If $0<k<1$, multiplying by $k$ at each step makes the outputs smaller, so the graph falls from left to right. For example, the outputs of $y=(1/2)^x$ halve at each integer step. In this case, the curve approaches zero as increases.

In both cases $y=0$ is a horizontal asymptote: the curve approaches the x-axis but never meets it. The special case $k=1$ gives $y=1^x=1$, so its graph is the horizontal line rather than a growth or decay curve.

Exponential change in context

Exponential change occurs when a quantity is repeatedly multiplied by the same factor. A 50% daily increase uses a multiplier of $1.5$, so a population starting at 1,000 follows $1000(1.5)^{n-1}$ at the start of day $n$. The day-by-day values are discrete, but they lie on the corresponding increasing exponential curve. A larger growth multiplier produces faster growth: populations beginning at the same value but growing by 50% and 30% per day initially share a point, after which the 50% curve rises more steeply.

Recognising and interpreting the two shapes

The reciprocal graph $y=1/x$ has two branches, no intercepts and two asymptotes: $x=0$ and . The exponential graph is one continuous curve, crosses the y-axis at and has only the horizontal asymptote .

To interpret either graph, read a coordinate as an input-output pair. On $y=1/x$, $(4,1/4)$ means that the reciprocal of 4 is $1/4$. On $y=2^x$, means . The direction and steepness of an exponential curve also show whether repeated multiplication is producing growth or decay.

Reciprocal graph

For $y=1/x$:

• $x\ne0$ and $y\ne0$;