A quadratic function is a function of the form

f(x) = ax² + bx + c (with the coefficient "a" nonzero).

Its graph is a parabola.

The vertex of this parabola occurs at the point on the graph with x coordinate -b/(2a).

It crosses the y-axis (y-intercept) at y = c.

It crosses the x-axis (x-intercept(s)) at the solutions of the quadratic equation ax² + bx + c = 0 (if there are any).

It is symmetric around the vertical line through the vertex.

If the coefficient (a) of x² is positive, it is concave up (as in the example to the right). If a is negative, it is concave down (as in the figure below).

The parabola

y = x² - 2x - 8

-

b 2a

=

2 2

=

1.

y = (1)² -2(1) - 8 = -9.

x² - 2x - 8 = 0
(x + 2)(x - 4) = 0,

An exponential function is a function of the form

f(x) = Ab^x,

The function f(x) = 3(2^x) is an exponential function with A = 3 and b = 2. It has the following graph.

The following table shows the y-coordinates of points on this graph. All you do is supply the x-coordinates and press "Compute y"

If b and c are positive, and x and y are any real numbers, then the following laws hold.

Law	Example
bxby = bx+y	2322 = 25 = 32
bx by = bx-y	43 42 = 41 = 4
1 bx = b-x	1 90.5 = 9-0.5 = 1 3
b0 = 1	(3.3)0 = 1
(bx)y = bxy	(32)2 = 34 = 81
(bc)x = bx cx	(4 2)2 = 4222 = 64
b c x = bx cx	4 3 2 = 42 32 = 16 9

Future Value
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

A

=

P

1

+

r m

mt

A(t)

=

P

1

+

r m

mt

You invest $1,000 at an annual rate of 4.8% interest, compounded monthly. This means that

P = 1,000,     r = 0.048,     m = 12.

A(t)	=	1,000 1 + 0.048 12 12t
	=	1,000(1.004)12t	.

A(5) = 1,000(1.004)¹²⁵ = 1,000(1.004) ⁶⁰ = $1,270.64.

The numbers

1

+

1 m

m

m	1	10	100	1000	10000
1 + 1 m m	2	2.59374246	2.70481383	2.71692393	2.71814593

The number e appears in the formula for continuous compounding: If $P is invested at an annual interest rate r compounded continuously, then the accumulated amount after t years is

A = Pe^rt.

r_e = e^r - 1.

If $1,000 is invested at an annual interest rate of 4.8% compounded continuously, then the accumulated amount after t years is

A = 1,000e^0.048t.

r_e = e^0.048 - 1 0.04917,

The statement

log_ax = y

a^y = x.

The following table lists some exponential equations and their equivalent logarithmic form.

The following identities hold for any positive a 1 and any positive numbers x and y.

Identity	Example
loga(xy) = logax + logay	log216 = log28 + log22
loga(x/y) = logax - logay	log2 (5/3) = log25 - log23
loga(xr) = r logax	log2(65) = 5 log26
logaa = 1 loga1 = 0	log22 = 1 log31 = 0
loga(1/x) = -logax	log2(1/3)= -log23
logax = log x log a = ln x ln a	log25 = log 5 log 2 2.3219

If a is any positive number, then the functions f(x) = log_ax and g(x) = a^x are inverse functions. This means that

a^logax= x

log_a(a^x) = x

Want to learn more about inverse functions? Go to our on-line text on inverse functions.

2^log2x= x

e^{ln x}= x

log₂(2^x) = x

ln (e^x) = x

A logistic function has the form

f(x) =

N 1 + Ab-x

A, N, b constant (b positive and 1)

Properties of the Logistic Curve

• The graph is an S-shaped curve sandwiched between the horizontal lines y = 0 and y = N. N is called the limiting value of the logistic curve.
• If b > 1 the graph rises; if b < 1, the graph falls.
• The y intercept is N/(1 + A)
• Role of b: For small x, the logistic function grows approximately exponentially with base b, and follows the curve [N/(1+A)] b^x.

b > 1

0 < b < 1

N = 50, A = 24, b = 3 gives

f(x) = 50 1 + 24(3-x)

Technology format: 50/(1+24*3^(-x))

The following figure shows the graph of f together with the exponential approximation

Logistic curve: 50/(1+24*3^(-x))
Exponential curve: 2*3^x