Topic 2.3 Notes – Exponential Functions

What an Exponential Function Is

The general form is

$$f(x)=ab^x$$

with $a\neq 0$, $b>0$, and $b\neq 1$.

A few pieces matter right away:

• The variable is in the exponent, so this is different from a power function like $x^2$.
• $a$ is the initial value because $$f(0)=ab^0=a$$ so the $y$-intercept is $(0,a)$.
• $b$ is the multiplicative factor for a 1-unit increase in $x$: $$f(x+1)=bf(x)$$

More generally, equal input intervals give equal output ratios:

$$\frac{f(x+h)}{f(x)}=b^h$$

That constant-ratio idea is the whole topic. Linear functions have constant differences. Exponential functions have constant ratios.

For natural-number inputs, the exponent tells you how many factors of $b$ get applied to $a$.
Example: $f(x)=3(2)^x$

• $f(0)=3$
• $f(1)=3(2)$
• $f(2)=3(2)(2)=12$

The domain is all real numbers, so the graph is a smooth continuous curve, not separate dots like a geometric sequence.

Why the restrictions matter:

• $a=0$ gives the zero function
• $b=1$ gives a constant function
• $b=0$ or $b<0$ fails to give real outputs for all real $x$

Key Characteristics from $a$ and $b$

When you analyze one, check this full set: growth or decay, intercepts, domain/range, increasing or decreasing, concavity, asymptote, end behavior, extrema.

This set of graphs puts all four sign-and-base cases on the same axes, so you can compare how $a$ and $b$ change the shape.

• If $a>0$ and $b>1$, it shows exponential growth.
• If $a>0$ and $0<b<1$, it shows exponential decay.

Intercepts domain and range

• Domain is $(-\infty,\infty)$
• $y$-intercept is $(0,a)$
• No $x$-intercept since $ab^x\neq 0$
• Range is $(0,\infty)$ if $a>0$, and $(-\infty,0)$ if $a<0$

Increasing decreasing concavity extrema

The base alone is not enough. The sign of $a$ matters too.

Concavity:

• $a>0$ means concave up
• $a<0$ means concave down
• No inflection points
• No local or absolute extrema on all real numbers
• On a closed interval, extrema can happen at endpoints

Asymptote and end behavior

The horizontal asymptote is always $y=0$.

• If $b>1$, the graph approaches $0$ on the left
• If $0<b<1$, the graph approaches $0$ on the right
• It approaches from above when $a>0$, from below when $a<0$

Recognizing Exponential Behavior in Equations Tables and Graphs

From an equation, look for a constant times a positive constant base raised to a variable exponent.

From a table, check ratios over equal input steps, not differences.

• Linear example idea: $+4,+4,+4$
• Exponential example idea: $\times 2,\times 2,\times 2$

If inputs go $0,2,4,6$, the equal step is 2, so the constant ratio is $b^2$, not $b$.

From a graph, look for:

• a smooth curve
• entirely above or entirely below the $x$-axis
• passing through $(0,a)$
• always increasing or always decreasing
• only one concavity
• approaching $y=0$ in one direction

Additive Transformations of Exponential Functions

A vertical shift gives

$$F(x)=ab^x+k$$

Now the outputs usually do not have a constant ratio. But if you remove the shift, exponential behavior appears:

$$F(x)-k=ab^x$$

Example with outputs $17,29,53,101$:

1. Raw ratios are not constant.
2. Subtract 5 from each output.
3. You get $12,24,48,96$.
4. Those have constant ratio 2.

So

$$F(x)=12(2)^x+5$$

That shift changes:

• asymptote to $y=k$
• range to $(k,\infty)$ if $a>0$, or $(-\infty,k)$ if $a<0$
• $y$-intercept to $F(0)=a+k$

It does not change:

• domain
• increasing/decreasing pattern
• concavity
• absence of inflection points

Common Mistakes to Catch Fast

• Constant differences do not mean exponential.
• Equal ratios only count when the input intervals are equal.
• $b<1$ does not automatically mean decreasing. Check the sign of $a$.
• Exponential functions in general form do not cross the $x$-axis.
• Only one end approaches the horizontal asymptote.
• In $ab^x+k$, the asymptote is $y=k$, not $y=0$.
• In $ab^x+k$, the $y$-intercept is $a+k$, not just $a$.