Exponentials and their Graphs

Introduction to the Exponential Graph

The exponential function $y = a^x$ is distinct because the variable $x$ appears as the exponent. This creates a specific family of curves. For the TMUA and ESAT, we focus on cases where the base $a$ is a simple positive value.

A crucial feature of all such graphs is that they intersect the y-axis at the point $(0, 1)$. This occurs because $a^0 = 1$ for any positive value of $a$. Furthermore, because $a$ is positive, $a^x$ will always be positive regardless of the value of $x$. This means the graph stays entirely above the x-axis, which acts as a horizontal asymptote.

The Three Cases for the Base $a$

The shape of the graph $y = a^x$ depends entirely on the value of the base $a$. We categorise these into three distinct cases.

Case 1: $a > 1$ (Exponential Growth)

When the base is greater than 1, the function represents exponential growth. As $x$ increases, $y$ increases at an accelerating rate. As $x$ becomes more negative, the value of $y$ gets closer and closer to 0 but never reaches it. For example, if we consider $y = 2^x$, as $x$ moves from 1 to 2 to 3, $y$ moves from 2 to 4 to 8. Conversely, as $x$ moves to -1, -2, and -3, $y$ becomes $1/2$, $1/4$, and $1/8$.

Case 2: $0 < a < 1$ (Exponential Decay)

When the base is between 0 and 1, the function represents exponential decay. This graph is effectively a reflection of a growth graph in the y-axis. For instance, $y = (1/2)^x$ is identical to $y = 2^{-x}$. As $x$ increases, the value of $y$ gets smaller, approaching the x-axis as an asymptote. As $x$ becomes more negative, $y$ grows very large.

Case 3: $a = 1$

This is a trivial or boundary case. Since $1$ raised to any power is always 1, the equation $y = 1^x$ simplifies to the horizontal line $y = 1$. It does not exhibit the characteristic growth or decay curve of other exponential functions.

Why must $a$ be positive?

The specification restricts our study to positive values of $a$. This is because negative bases lead to complex mathematical issues. Consider $a = -64$: if $x = 1/3$, then $(-64)^{1/3} = -4$, which is a real number. However, if $x = 1/2$, then $(-64)^{1/2}$ is the square root of a negative number, which does not exist in the real number system.

If we attempted to graph $y = (-64)^x$, we would see a series of disconnected dots rather than a smooth, continuous curve. To maintain a functional, continuous curve that behaves predictably across all real values of $x$, we require $a > 0$.

The Effect of Increasing the Base

You should understand how the graph changes as $a$ gets larger (assuming $a > 1$). As $a$ increases:

1. For positive values of $x$, the graph becomes steeper, rising much faster toward infinity.
2. For negative values of $x$, the graph approaches the x-axis ($y = 0$) more rapidly.
3. The graph still passes through $(0, 1)$ regardless of how large $a$ becomes.

By comparing $y = 2^x$ and $y = 10^x$, you can observe that $10^x$ is 'higher' than $2^x$ for all $x > 0$, but 'lower' (closer to the axis) for all $x < 0$.