The exponential function

In an exponential function the variable is in the exponent, and the base is a constant number:

• A negative base has no value for some $\displaystyle x$: $\displaystyle {(-9)^{\frac{1}{2}}}={\sqrt{-9}}$ is not a number.
• The base 1 gives $\displaystyle {1^x}={1}$ for every $\displaystyle x$ – a horizontal line.

On the axes: $\displaystyle 3^x$ and $\displaystyle \left(\frac{1}{3}\right)^x$.

Every exponential function $\displaystyle {y}={a^x}$ has the same properties:

• The domain: every $\displaystyle x$.
• $\displaystyle a^x>0$ for every $\displaystyle x$: the graph does not meet the x-axis.
• It meets the y-axis at $\displaystyle (0,1)$, because $\displaystyle {a^0}={1}$.
• The horizontal asymptote $\displaystyle {y}={0}$: for $\displaystyle a>1$ the graph approaches it on the left, for $\displaystyle 0<a<1$ on the right.
• For $\displaystyle a>1$ the function increases; for $\displaystyle 0<a<1$ it decreases.

$\displaystyle {\left(\frac{1}{3}\right)^x}={3^{-x}}$: at $\displaystyle x$ it has the value $\displaystyle 3^x$ takes at $\displaystyle -x$.

• So the graph of $\displaystyle \left(\frac{1}{3}\right)^x$ is the mirror image of the graph of $\displaystyle 3^x$ in the y-axis.
• The same holds for every base: $\displaystyle {\left(\frac{1}{a}\right)^x}={a^{-x}}$.

In an exponential function, each step of 1 in $\displaystyle x$ multiplies $\displaystyle y$ by the same number. In a linear function each such step adds the same number to $\displaystyle y$:

• The values of $\displaystyle f$ are a geometric sequence with the ratio 2, the base.
• The values of $\displaystyle g$ are an arithmetic sequence with the difference 2, the slope.
• So a constant ratio of consecutive values, at equal steps of $\displaystyle x$, shows that the function is exponential.

In the function $\displaystyle {y}={k\cdot a^{x-p}+q}$, the graph of $\displaystyle a^x$ moves $\displaystyle p$ to the right and $\displaystyle q$ up, and is stretched by a factor of $\displaystyle k$. The asymptote moves with the graph, to $\displaystyle {y}={q}$.
The example below finds the properties of $\displaystyle {f(x)}={2^{x-1}-4}$:

• $\displaystyle 2^x$ shifted 1 to the right and 4 down; the asymptote is $\displaystyle {y}={-4}$.
• The x-axis: $\displaystyle {2^{x-1}}={4}={2^{2}}$, so $\displaystyle {x-1}={2}$ and $\displaystyle {x}={3}$.
• The y-axis: $\displaystyle {f(0)}={2^{-1}-4}={-3.5}$.
• The base 2 is greater than 1, so the function increases.

The function $\displaystyle {f(x)}={3^{x-2}-1}$ is given.

Write the equation of the horizontal asymptote, and decide whether the function increases or decreases.

Find the point where the graph meets the x-axis.

Mark on the graph the points where it meets the axes.

Find the function, and write it in a row of your own.

The function is of the form $\displaystyle {y}={a^{x}+q}$.
The asymptote of its graph is $\displaystyle {y}={-2}$, and the graph passes through the point $\displaystyle A(1,1)$.

The function is of the form $\displaystyle {y}={k\cdot a^{x}}$.
Its graph passes through the points $\displaystyle A(0,5)$ and $\displaystyle B(2,20)$.

The function $\displaystyle {f(x)}={k\cdot a^{x}}$ takes the values:
$\displaystyle {f(1)}={6}$, $\displaystyle {f(2)}={18}$, $\displaystyle {f(3)}={54}$.

Find $\displaystyle a$ and $\displaystyle k$, and write the function in a row of your own.

Calculate $\displaystyle f(5)$.

The functions $\displaystyle {f(x)}={2^{x}}$ and $\displaystyle {g(x)}={8\cdot\left(\frac{1}{2}\right)^{x}}$ are given.

Find the x-coordinate of the point where the graphs meet.

Mark the meeting point on the graph.

Find the values of $\displaystyle x$ for which $\displaystyle f(x)>g(x)$.