Transcript Chapter 1

CHAPTER 1
College
Algebra
GRAPHS
 Graphs provide a means of displaying, interpreting and
analyzing data in a visual format.
 We use a plane to graph a pair of numbers
 To locate points on a plane, we use two perpendicular number
lines called axes
 The two axes intersect at (0,0).
 The origin
 The horizontal axis is called the x -axis and the vertical axis is
called the y -axis
 The axes divide the plane into four regions called quadrants
denoted by Roman numerals and numbered counterclockwise
from the upper right.
GRAPHS
 Each point (x,y) in the plane is called an ordered pair. The first
number, x, indicates the point’s horizontal location with
respect to the y -axis and the second number, y, indicates the
points vertical location with respect to the x -axis.
 We call x the first coordinate, x-coordinate, or abscissa
 We call y the second coordinate, y -coordinate, or ordinate
 Such a representation is called the Cartesian coordinate
system in honor of the French mathematician and philosopher
Rene Descartes.
 In the first quadrant, both coordinates are positive.
 What about the second quadrant?
 The third?
 And fourth?
SOLUTIONS OF EQUATIONS
 Equations in two variables, like 2x + 3y = 18, have solutions
(x,y) that are ordered pairs such that when the first coordinate
is substituted for x and the second coordinate is substituted
for y, the result is a true equation, The first coordinate in an
ordered pair generally represents the variable that occurs first
alphabetically.
 Determine whether each ordered pair is a solution of the equation
mentioned above:
 (-5,7)
 (3,4)
GRAPHS OF EQUATIONS
 The equation that we just looked at has an infinite number of
solutions
 Although we can not list all of the solutions, we can create a
graph, a drawing, that represents them.
 To graph an equation is to make a drawing that represents the
solutions of that equation
 An x-intercept is a point (a, 0). To find a, let y = 0 and solve
for x.
 A y -intercept is a point (0,b). To find b, let x = 0 and solve for
y.
 Graphs of equations of the type Ax + By = C are straight lines
SUGGESTIONS FOR GRAPHING AN
EQUATION
 1 . Calculate solutions and list the ordered pairs in a table
 2. Use graph paper
 In this class, you will ALWAYS be expected to use graph paper when
drawing a graph.
 3. Draw axes and label them with the variables
 4. Use arrows on the axes to indicate positive direction
 5. Scale the axes; that is, label the tick marks on the axes.
Consider the ordered pairs when choosing a scale
 6. Plot the ordered pairs, look for patterns, and complete the
graph. Label the graph with the equation that is being
graphed.
WHEN GRAPHING
 Sometimes it is easier to first solve for y and then find the
ordered pairs
 We can use addition and multiplication principles to solve for
y.
 Graph: 3x – 5y = -10
 First sole for y
 y = 3x/5 + 2
 The value of y depends on the value chosen for x, so is said to
be the independent variable and y the dependent variable
USING A CALCULATOR
 You can also graph equations using a calculator.
 When using a calculator, the equation must be written
explicitly or in the form of y =
 You also must be aware of the viewing window.
 If there are any questions about graphing with a calculator we
will do an example as a class using a TI -83 graphing
calculator.
 If not, then we will move on.
 Please note that you will very rarely if ever use a calculator
during this course.
THE DISTANCE FORMULA
 The strategy used to find the distance between two points in a
plane is to use the Pythagorean Theorem
 For two points (a,b) and (c,d), we can draw a right triangle in
which the legs have lengths a -c and b-d
 (remember that with distances, you use absolute value)
 The distance d between any two points ( a,b) and (c,d) is given
by
d=
(a - c) + (b - d )
2
2
THE MIDPOINT FORMULA
 The distance formula can be used to develop a way of
determining the midpoint of a segment when the endpoints
are known:
 Midpoint: If the endpoints of a segment are ( a,b) and (c,d)
then the coordinates of the midpoint are:
a+c b+d
(
,
)
2
2
CIRCLES
 A circle is a set of all points in a place that are a fixed
distance r from a center ( h,k).
 The equation for a circle is as follows
 With a center (h,k) and a radius r
(x-h) +(y-k) =r
2
2
2
1.2 FUNCTIONS AND THEIR GRAPHS
 A function is a correspondence between a first set, called the
domain, and a second set, called the range, such that each
member of the domain corresponds to exactly one member of
the range.
 It is important to note that not every correspondence between two
sets is a function.
 A relation is a correspondence between the first set, called
the domain, and a second set, called the range, such that
each member of the domain corresponds to at least one
member of the range.
 Relations are important to note due to the fact that a
correspondence can be consider a relation even if it is not considered
a function
NOTATION
 A concise notation is often used with functions.
 The inputs (members of the domain) are values of x
substituted into the equation.
 The outputs (members of the range) are the resulting values
of y,
 If we call the function f, we can use x to represent an arbitrary
input and f(x) to represent the corresponding output.
 Read as f of x
 Or as f at x
GRAPHS OF FUNCTIONS
 We graph functions the same way that we graph equations:
 We find ordered pairs (x, y) or (x, f(x)), plot points and then
complete the graph.
 We know that when one member of the domain is paired with
two or more dif ferent members of the range, the
correspondence is not a function. Thus, when a graph contains
two or more dif ferent points with the same first coordinate,
the graph cannot represent a function, Points sharing a
common first coordinate are vertically above or below each
other.
 The vertical line test
 If it is possible for a vertical line to cross a graph more than once,
then the graph is not a graph of a function
FINDING DOMAINS OF FUNCTIONS
 When a function f, whose inputs and outputs are real
numbers, is given by a formula, the domain is understood to
be the set of all inputs for which the expression is defined as
a real number.
 When a substitution results in an expression that is not
defined as a real number, we say that the function value does
not exist and that the number being substituted is not in the
domain of the function.
 Example: Find the domain for f(x) = 1/(x -3)
 Since division by zero is not defined, 3 is not in the domain.
 The domain is
{x | x ¹ 0}
 Or it can be written as (-¥, 3)È(3, ¥)
 Where the u represents the term union, or inclusion of both sets.
VISUALIZING DOMAIN AND RANGE
 Keep the following in mind regarding the graph of a function:
 Domain = the set of a function’s inputs, found on the horizontal axis
 Range = the set of a function’s outputs, found on the vertical axis
 Always consider adding the reasoning “What can I
substitute?” to find the domain.
 Think “What do I get out?” to find the range.
HOMEWORK
1.3 LINEAR FUNCTIONS, SLOPE AND
APPLICATIONS
 When given information, if we use that information to
formulate an equation or inequality that at least approximates
the situation mathematically then we have created a model.
 The most frequently used mathematical model is graphical modeling
 There is also algebraic modeling and modeling with a table
 One of the most frequently used graphical models is linear –
the graph of a linear model is a straight line
 Linear functions
 A function f is a linear function if it can be written as
f(x) = mx + b
where m and b are constants.
 If m = 0, the function is the constant function f(x) = b. If m = 1 and b
= 0, the function is the identity function f(x) = x.
TWO T YPES OF LINES
 Horizontal lines are given by equations of the type y = b or f(x)
= b
 Horizontal lines ARE functions
 Vertical lines are given by equations of the type x = a
 Vertical lines are NOT functions
SLOPE
 The slope m of a line containing points ( a,b) and (c,d) is given
by




Rise/run
The change in y / the change in x
Average rate of change
(d – b) / (c – a)
 It important to remember that it doesn’t matter which point is first as long
as you are consisting on the numerator & the denominator
 The slope of the line of f(x) = mx + b is m
 Horizontal lines – the change in y for any two points is zero
and the change in x is nonzero, thus the slope is zero.
 Vertical lines – the change in y for any two points is nonzero
and the change in x is zero, thus the slope is undefined.
1.4 EQUATIONS OF LINES AND MODELING
 Slope-Intercept Equations of Lines
 y = mx + b
 Where we know m is the slope and b is the y -intercept
 The linear functions f is given by f(x) = mx + b
is written in slope -intercept form. The graph of an equation in
this form is a straight line parallel to y = mx: The constant m
is called the slope, and the y -intercept is (0,b)
 The point-slope equation:
 The line with slope m passing through the point ( a,b) is
y – b = m (x – a)
EQUATIONS OF LINES
 Parallel Lines
 Vertical lines are parallel. Nonvertical lines are parallel if and only if
they have the same slope and different y -intercepts.
 Perpendicular Lines
 Two lines with slopes m 1 and m 2 are perpendicular if and only if the
product of their slopes is -1:
m 1 m 2 = -1
 Lines are also perpendicular is one is vertical (x = a) and the other is
horizontal (y = b).
MATHEMATICAL MODELS
 When a real-world problem can be described in mathematical
language, we have a mathematical model. Situations in which
algebra can be brought to bear often require the use of functions
as models.
 Creating a mathematical model





Recognize real-world problem
Collect data
Plot data
Construct model
Explain and predict
 Curve Fitting
 Functions that can be used to model data include linear, quadratic,
cubic, and exponential.
 In general we try to find a function that fits the data as well as possible,
observations(data), theoretical reasoning, and common sense – we call
this curve fitting.
MODELING
 Scatterplots are a simple way to find trends in sets of data.
 They are created by simply plotting the points.
 Using a calculator, we can find a model to fit the data of a
scatterplot using the regression function.
 In particular, we will currently be using linear regression which is a
procedure used to model a set of data using a linear function.
 Linear correlation coef ficient – a constant r, a real number
between -1 and 1 , which is used to describe the strength of
the linear relationship between x and y.
 The closer the absolute value of r is to 1, the better the correlation
 A positive r indicates that the regression line has a positive slope,
and a negative value of r indicates that the regression line has a
negative slope.
HOMEWORK
1.5 MORE ON FUNCTIONS
 Increasing functions – if a graph rises from left to right it is
said to be increasing on that interval.
 A function f is said to be increasing on an open interval I, for all a
and b in the in that interval, a < b implies f(a) < f(b).
 Decreasing functions – if the graph drops from left to right, it
is said to be decreasing.
 A function f is said to be decreasing on an open interval I, for all a
and b in that interval, a < b implies f(a) > f(b).
 Constant functions – if a function value stays the same from
left to right then it is said to be constant.
 A function f is said to be constant on an open interval I, if for all a
and b in that interval f(a) = f(b).
RELATIVE MAX AND MIN VALUES
 Suppose that f is a function for which f(c) exists for some c in
the domain of f. Then:
 f(c) is a relative maximum if there exists an open interval I containing
c such that f(c) > f(x), for all x in I where x does not equal c
 f(c) is a relative minimum if there exists an open interval I containing
c such that f(c) < f(x) for all x in I where x does not equal c
 Simply stated, f(c) is a relative maximum if f(c) is the highest
point, “peak”, in some open interval and f(c) is a relative
minimum if f(c) is the lowest point, “valley”. in some open
interval.
PIECEWISE FUNCTIONS
 Sometimes functions are defined as piecewise using dif ferent
output formulas for dif ferent parts of the domain.
 The example on the right shows
what a piecewise graph looks
like when it is written out and
what it looks like to graph.
 This one is particular, shows
two graphs that meet at
the point where they split.
This will not always be the case.
GREATEST INTEGER FUNCTION
1.6 THE ALGEBRA OF FUNCTIONS
 Sums, Dif ferences, Products and Quotients of Functions
 If f and g are functions and x is in the domain of each
function, then




(f+g)(x) = f(x) + g(x)
(f-g)(x) = f(x) – g(x)
(fg)(x) = f(x)g(x)
(f/g)(x) = f(x) / g(x) provided that g(x) does not equal zero.
 Domains:
 If f and g are functions, then the domain of the functions f+g, f-g, and
fg are each the intersection of the domain of f and the domain of g.
The domain of f/g is also the intersection of the domains of f and g
with the exclusion of any x-values for which g(x) = 0.
DIFFERENCE QUOTIENTS
 We know that the slope of a line is the average rate of change
of that line.
 Let’s consider a nonlinear functions f and draw a line through
two points – the slope of the line is called the secant line in
this case.
 The ratio is called the dif ference quotient or the average rate
of change.
f (x + h) - f (x)
h
COMPOSITION OF FUNCTIONS
 The composition of f and g is defined as
f(g(x)) = (f
g)(x)
where x is in the domain of g and g(x) is in the domain of f.
 Example:
 Given that f(x) = 2x + 6 and g(x) = x 2 +x+3. Find f(g(x)), g(f(x))
and f(g(3))
 f(g(x)) = 2(x 2 +x+3) + 6 = 2x 2 + 2x + 6 + 6 = 2x 2 + 2x + 12
 g(f(x)) = (2x+6) 2 + 2x+6 + 3 = 4x 2 + 12x +12x + 36 + 2x + 6 + 3
= 4x 2 + 26x + 45
 f(g(3)) = [3 2 +3+3 = 15] so 2(15) + 6 = 36
1.7 SYMMETRY AND TRANSFORMATIONS
 Symmetry
 Symmetry often occurs in nature and in art. A knowledge of
symmetry in mathematics helps us graph and analyze equations and
functions.
 Points that have the same x-value but opposite y -values are
reflections of each other across the x-axis,
 If for any point (x,y) on a graph, the point (x, -y) is also on the graph then
the graph is said the be symmetric with respect to the x -axis.
 Points can also be reflected about the y -axis
 If for any point (x,y) on a graph, the point (-x, y) is also on the graph then
the graph is said to be symmetric with respect to the y -axis
 Finally, you can also discuss symmetry as a reflection about the
origin
 If for any point (x,y) on a graph, the point (-x, -y) is also on the graph then
the graph is said to be symmetric with respect to the origin,
ALGEBRAIC TEST OF SYMMETRY
 x-axis: If
then the
 y -axis: If
then the
replacing y with –y produces an equivalent equation,
graph is symmetric with respect to the x -axis
replacing x with –x produces an equivalent equation,
graph is symmetric with respect to the y -axis
 If a function f is symmetric with respect to the y -axis, we say that it is
an even function. That is for each x in the domain of f, f(x) = f( -x)
 origin: if replacing x with –x and y with –y produces an
equivalent equation, then the graph is symmetric with respect
to the origin.
 If the graph of a function f is symmetric with respect to the origin, we
say that it is an odd function. That is, for each x in the domain of f,
f(-x) = -f(x)
 With the exception of the function f(x) = 0, a function can not
be both odd and even at the same time.
BASIC GRAPHS
TRANSLATIONS
 Horizontal translations: y = f(x plus/minus d)
 For d > 0
 The graph of y = f(x –d) is the graph of y = f(x) shifted right d units
 The graph of y = f(x + d) is the graph of y = f(x) shifted left d units
 Vertical translations: y = f(x) plus or minus b
 For b > 0
 The graph of y = f(x) + b is the graph of y = f(x) shifted up b units
 The graph of y = f(x) – b is the graph of y = f(x) shifted down b units
 Reflections
 Across the x-axis: the graph of y = -f(x) is the reflection of the graph
of y = f(x) across the x-axis
 Across the y-axis: the graph of y = f(-x) is the reflection of the graph
of y = f(x) across the y -axis
TRANSFORMATIONS
TRANSFORMATIONS
HOMEWORK