Ch. 1: Investigations & Functions . Functions
• Graphs w/ Asymptoles
• Polynomials o Laws of Exponents . Solving Quadratics
• Describing Spread
• Describing Data Distribution
Functions, Graphs w/ Asymptotes, Polynomials.
Laws of Exponents
Functions
A relationship between inputs and outputs is a function if there is exactly one output for each
input. Functions are often written as y =(some expression involving .x), where x is the input and
y is the output. The following equation, table, and graph represent the same function.
y =(x - 2)2
ht
-2
-1
0
1
2
3
4
5
HAH,
5
st
The set of possible input values for a function is called the domain. This set contains
every x-value for which the function is defined. The domain of the function above is all
real numbers.
The range of a function is the set of possible output values. This set contains every
y-value that the function can generate. As the graph shows, the minimum value of the
function occurs at the vertex, where y = 0. So the range of the function above is y 0 .
Another way to write a function is with the notation W/(x) = " instead of 1=57 "y=". The function
named "fhas output f(x). The input is x.
In the example at right. A5)=9. The input is 5 and the output is 9. You read this as, "fof 5 equals
9."
f(x) = (1-2)
x2 + y2 = 1 is not a function because there are two y-values (outputs) for
some r-values, as shown below.
|(5) =
32 + 12 = 1
- y 0110
DT
I
Graphs with Asymptotes
A mathematically clear and complete definition of an asympto calculus, but
some examples of graphs with asymptotes should help when they occur. In
the following examples, the dotted lines are the equations of the
asyimptotes are given. In the two lower grante asymptote.
of an asymptote requires some ideas from
tes should help you recognize them ted lines are the asymptotes, and the
ower graphs, the v-axis, x = 0, is also an
#3
y=5
1
x = 2
na
?
*
asymptotes
As you can see in the examples, asymptotes can be diagonal lines or even
curves. However, in this course, asymptotes will almost always be
horizontal or vertical lines. The graph of a function has a horizontal
asymptote if, as you trace along the graph out to the left or right (that is, as
you choose x-values farther and farther away from zero, either toward
negative infinity or toward positive infinity), the distance between the graph
of the function and the asymptote gets smaller and smaller.
A graph has a vertical asymptote if, as you choose .r-coordinates closer
and close to certain value, from either the left or right (or both), the
p-coordinate gets farther away zero. either toward infinity or toward
negative infinity.
rom
Polynomials
A polynomial expression in one variable is an expression that can be written as the sum
or difference of terms of the form:
(any real number)x(whole number)
Polynomials in one variable (often .x) are usually arranged with powers of x
in order, starting with the highest, from left to right.
The highest power of the variable in a polynomial in one variable is called the degree of the
polynomial. The numbers that are multiplied by the variable in each term are called coefficients.
These descriptions also apply to functions defined by polynomial expressions. See the
examples below.
Example 1: 3r + 5 is a polynomial of degree 1 with coefficients 3 and 5.
Note that the last term, 5, is called the constant term but represents the
expression 5.ro. since.ro = 1. The related linear function 1: = 3x + 5 is a
polynomial function of degree 1.
Example 2: 77% +2.5m2 - 4x + 7 is a polynomial of degree 5 with
coefficients 7, 0,2.5.0. - 1. and 7.
Example 3: 2(x + 2)(x + 5) is a polynomial in factored form with degree 2
because it can be written in standard form as 2x2 + 14x + 20. The standard
form has coefficients 2. 14. and 20. The related quadratic function f(x) =
2x2 + 14.2 + 20 is a polynomial function of degree 2.
Laws of Exponents
In the expression r. r is the base and 3 is the exponent. Here are the laws
of exponents with examples:
Law
Examples
x x = x+for all x hty = xm- for *#0
*-*** = * *4 = ?
0 +* = 410-4 = x6
(x) = xm
for all.x
= x+3 72
28.2-1 = 24
= 5-} (10') = 10
90 = 1
3-1 = } 8273 - 182 - 164 = 4
x0=1 for x70
*"- for x#0 -2xmin - "x" for n #0 -4/5 - Vx+
Solving Quadratics, Describing Spread,
Describing Data Distribution Solving a
Quadratic Equation
In a previous course, you learned how to solve quadratic equations (equations that can be
written in the form ar? + bx+c=0). Review two methods for solving quadratic equations below.
Some quadratic equations can be solved by factoring and then using thie Zero Product
Property. For example, the quadratic equation r2 – 3r - 10 = 0 can be rewritten by
factoring as (x5)(x + 2) = 0. The Zero Product Property states that if ab = 0. then a=0 or
b=0. So if (x - 5)(x + 2) = 0. then (x - 5) = 0) or (r + 2) = 0. Therefore, r = 5 ory=-2.
Another method for solving quadratic equations is using the Quadratic Formula. This
method is useful for solving quadratic equations that are difficult or impossible to factor.
Before using the Quadratic Formula, shown below, the quadratic equation you want to
solve must be in standard forin (that is, written as ar? + hr+c=0).
.
ax #bX+1:0,
- bV62 - 4ac
2a
The Quadratic Formula gives two possible solutions for r. The two solutions are shown
by the "3" symbol. This symbol (read as "plus or minus”) is shorthand notation that tells
you to evaluate the expression twice: once using addition and once using subtraction.
Therefore, Quadratic Formula problems usually must be simplified twice to give:
I=.
_ -b + Vb2 - 4ac
Le 2a
or r=
-- Vb2 - 400
2a
To solve r- - 3x - 10 = 0 using the Quadratic Formula, substitute a= 1, b = -3, and c=-10
into the formula, as shown below, then simplify.
X-
201)
-(-3)+/(-3)2-4(1)(-10)
2(1) r = 31749 t = or 근 x= 5 or -2
forens
Completing the square:
x?+ 4x + 10 =0 x?+4x4 =-10+4
(x+2) = -6 X+2 = +ive
X = -2* i Ja
Describing Spread
A distribution of single-variable data can be summarized by describing its center,
shape. spread awa outliers. Three ways to describe spread are explained below.
Range : max-min Range The range is the maximum minus the minimum. It is usually
not a good way to describe the sprea because it considers only the extreme values in
the data, rather than how the bulk of the data is spread. For the data set below, the
range is 30 - 11 = 19.
11,13,15,19 22,22,25,26,27,28,29) 29,30 min Q1 = 17 median Q3 = 28.5 max
Interquartile Range (IQR)
Q
The variability, or spread, in the distribution can be numerically summarized with the
interquartile range (IQR). The IQR is found by subtracting the first quartile (Q1) from the third
quartile (03). The IQR is the range of the middle half of the data. IQR can represent the spread
of any data distribution, even if the distribution is not symmetric or has outliers. In the data set
above, the IOR is Q3 - Q1 = 28.5 - 17 = 11.5.
Interquartile Range (IQR)
The variability, or spread, in the distribution can be mumerically summarized with the
interquartile range (IQR). The IQR is found by subtracting the first quartile (QI) from the
third quartile (Q3). The IQR is the range of the middle half of the data. IQR can
represent the spread of any data distribution, even if the distribution is not symmetric or
has outliers. In the data set above, the IQR is Q3 - Q1 = 28.5 - 17 = 11.5.
Standard Deviation
Either the interquartile range or standard deviation can be used to represent the spread if the
data is svinmetric and has no outliers. The population standard deviation is the square root of
the average of the distances to the mean, after the distances have been made positive by
squaring. For the data set above:
• There are 13 values in the population and the mean value is about 22.77.
• The sum of the squares of the distances from the mean is
(-11.77)2 + (-9.77)2 + (-7.77)2 + (-3.77)2 + (-0.77)2+(-0.77)2
+ 2.232 + 3.232 + 4.232 + 5.232 +6.232 – 500.3
• The population standard deviation i
500
6.2
Because gathering data for entire populations is often impractical, most of
the data sets we analyze are samples. To calculate the sample standard
deviation, divide by one less than the number of values in the data set. If
the data set above is a sample from a larger population, then the sample
standard deviation is 1.500.3 6.5 .
Describing a Data Distribution
Combination Histogram
and Boxplot
Single-variable numerical data can be represented graphically
with a combination histogram and boxplot. When describing
single-variable data, consider the:
• Center: mean or median
• Shape: symmetric, skewed, single- or double-peaked.
uniforin
He
100
2002
300
Woo
IMI muh mh mm
Symumetrie
skewed (left)
single peaked
cluble-Paked
unitorin
• Spread: standard deviation or interquartile range
• Outliers: data that is far from the bulk of the data
Shape Outlier Center Spread