It is symmetric around the vertical line through the vertex.
If the coefficient $a$ of $x^2$ is positive, it is concave up (as in the figure below when you press "$a > 0$"). If $a$ is negative, it is concave down (as in the figure below when you press "$a < 0$").

Examples: Quadratic functions and their graphs Quadratic function The function
Parabola The curve

$y=x^2-2x-8$

is a parabola with vertex at

$x = -\frac{b}{2a} = -\frac{-2}{2(1)} = 1.$

The $y$-coordinate of the vertex is

$y = (1)^2 - 2(1) - 8 = -9.$

The $y$-intercept is $c = -8$ and the $x$-intercepts are the solutions of

$x^2 - 2x - 8 = 0$ $(x+2)(x-4) = 0$,

so that $x = -2$ and $4$. Here is its graph:

Practice:

Laws of exponents
If $b$ and $c$ are positive, and $x$ and $y$ are any real numbers, then the following laws hold:

Exponential functions
An exponential function is a function of the form

$f(x) = Ab^x$,

where $A$ and $b$ are constants and $b \gt 0$.The number $b$ is called the base of the exponential function.Role of b

An increase of 1 unit in $x$ has the effect of multiplying $f(x)$ by $b$.

An increase of 2 units in $x$ has the effect of multiplying $f(x)$ by $b^2$.

...

In general, an increase of $\Delta x$ units in $x$ has the effect of multiplying $f(x)$ by $b^{\Delta x}$.

Role of A

$f(0) = A$, so $A$ is the $y$-intercept of the graph of $f$.

Examples: Exponential functions
The function $f(x) = 3(2^x)$ is an exponential function with $A = 3$ and $b = 2$. Some values of $f$ are shown in the following table:

Its graph is as follows:

Practice:

Compound interest
If an amount $P$ (the present value) earns interest at an annual interest rate $r$, compounded $m$ times per year, then the accumulated amount (or future value) after $t$ years is

$F(t) = P\left\[1+\frac{r}{m}\right\]^{mt}$.

This is an exponential function of $t$ of the form $F(t) = Ab^t$ as it can be written as

converge to the number
e = 2.71828182845904523536... as $m$ gets larger and larger. The following table shows the value of $\left(1+\frac{1}{m}\right)^m$ for several values of $m$.

Example: The number e
Enter your own value of $m$ and press "Compute". (You will notice that values of $m$ larger than around 100,000,000 will result in computational errors --- Experiment!).

$m = $

$\left(1+\frac{1}{m}\right)^m = $

Continuous compounding
The number $e$ appears in the formula for continuous compounding: If an amount $P$ (the present value) earns interest at an annual interest rate $r$, compounded continuously, then the accumulated amount (or future value) after $t$ years is

$F(t) = Pe^{rt}$.

The effective (annual) yield from continuous compounding is given by

$r_e = e^r - 1$.

Examples: Continuous compounding
If \$10,000 is invested at an annual interest rate of 4.8% compounded continuously, then the accumulated amount after $t$ years is

$A = 10\,000e^{0.048t}$

The effective annual yield is

$r_e = e^{0.048} - 1 \approx 0.04917$,

that is, 4.917% per year.

Logarithms
The statement:

$\log_bx = y$ The base b logarithm of x equals y

means

$b^y = x.$

Note:

$\log_{10}x$ is often written as $\log x$, and called the common logarithm of $x.$

The expression $log_ex$ is often written as $\ln x$ and called the natural logarithm of $x.$

Examples: Logarithms
The following table lists some exponential equations and their equivalent logarithmic form:

Practice:

Logarithm identities
The following identities hold for any positive base $b \neq 1$ and any positive numbers $x$ and $y.$