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PRECALCULUS I
Graphs and Lines
•Intercepts, symmetry, circles
•Slope, equations, parallel, perpendicular
Dr. Claude S. Moore
Danville Community College
1
Graph of an Equation
Equation - equality of two quantities.
Solution - (a,b) makes true statement when
a and b are substituted into equation.
Point-plotting method - simplest way to
graph.
x
-2 -1 0 1 2
y = 2x - 3 -7 -5 -3 -1 1
Finding Intercepts
of an Equation
The x-intercept is point where graph touches
(or crosses) the x-axis.
The y-intercept is point where graph touches
(or crosses) the y-axis.
1. To find x-intercepts, let y be zero and
solve the equation for x.
2. To find y-intercepts, let x be zero and
solve the equation for y.
Tests for Symmetry
1. The graph of an equation is symmetric with
respect to the y-axis if replacing x with -x yields
an equivalent equation.
2. The graph of an equation is symmetric with
respect to the x-axis if replacing y with -y yields
an equivalent equation.
3. The graph of an equation is symmetric with
respect to the origin if replacing x with -x and y
with -y yields an equivalent equation.
Standard Form of the
Equation of a Circle
The point (x, y) lies on the circle of
radius r and center (h, k)
if and only if
(x -
2
h)
+ (y -
2
k)
=
2
r
.
Slope-Intercept Form
of the Equation of a Line
The graph of the equation
y = mx + b
is a line whose slope is m and
whose y-intercept is (0, b).
Definition: Slope of a Line
The slope m of the nonvertical line
through (x1, y1) and (x2, y2)
where x1 is not equal to x2 is
y2  y1
m
x2  x1
Point-Slope Form
of the Equation of a Line
The equation of the line with
slope m passing through the
point (x1, y1) is
y - y1 = m(x - x1).
Equations of Lines
1. General form:
2. Vertical line:
3. Horizontal line:
4. Slope-intercept:
5. Point-slope:
1. Ax + By + C = 0
2. x = a
3. y = b
4. y = mx + b
5. y  y1 = m(x  x1)
Parallel and
Perpendicular Lines
Parallel: nonvertical l1 and l2 are
parallel iff m1 = m2 and b1  b2.
*Two vertical lines are parallel.
Perpendicular: l1 and l2 are
perpendicular iff
m1 = -1/m2 or m1 m2 = -1.
PRECALCULUS I
Functions and Graphs
•Function, domain, independent variable
•Graph, increasing/decreasing, even/odd
Dr. Claude S. Moore
Danville Community College
11
Definition: Function
A function f from set A to set B is a rule
of correspondence that assigns to each
element x in set A exactly one
element y in set B.
Set A is the domain (or set of inputs) of
the function f, and set B contains
range (or set of outputs).
Characteristics of a Function
1. Each element in A (domain) must be
matched with an element of B (range).
2. Each element in A is matched to not more
than one element in B.
3. Some elements in B may not be matched
with any element in A.
4. Two or more elements of A may be
matched with the same element of B.
Functional Notation
Read f(x) = 3x - 4 as “f of x equals three
times x subtract 4.”
x inside parenthesis is the
independent variable.
f outside parenthesis is the
dependent variable.
For the function f(x) = 3x - 4,
f(5) = 3(5) - 4 = 15 - 4 = 11, and
f(-2) = 3(-2) - 4 = - 6 - 4 = -10.
Piece-Wise Defined Function
A “piecewise function” defines the function
in pieces (or parts).
In the function below,
if x is less than or equal to zero,
f(x) = 2x - 1; otherwise, f(x) = x2 - 1.
 2 x  1 if x  0
f ( x)   2
 x  1 if x  0
Domain of a Function
Generally, the domain is implied to be the set
of all real numbers that yield a real
number functional value (in the range).
Some restrictions to domain:
1. Denominator cannot equal zero (0).
2. Radicand must be greater than or equal to
zero (0).
3. Practical problems may limit domain.
Summary of Functional Notation
In addition to working problems, you should
know and understand the definitions of
these words and phrases:
dependent variable
independent
variable
domain
range
function
functional notation
functional value
implied domain
Vertical Line Test for a Function
A set of points in a coordinate
plane is the graph of
y as a function of x
if and only if no vertical line
intersects the graph at more than
one point.
Increasing, Decreasing, and
Constant Function
On the interval containing x1 < x2,
1. f(x) is increasing if f(x1) < f(x2).
Graph of f(x) goes up to the right.
2. f(x) is decreasing if f(x1) > f(x2).
Graph of f(x) goes down to the right.
On any interval,
3. f(x) is constant if f(x1) = f(x2).
Graph of f(x) is horizontal.
Even and Odd Functions
1. A function given by y = f(x) is even if,
for each x in the domain,
f(-x) = f(x).
2. A function given by y = f(x) is odd if,
for each x in the domain,
f(-x) = - f(x).
PRECALCULUS I
Composite and
Inverse Functions
•Translation, combination, composite
•Inverse, vertical/horizontal line test
Dr. Claude S. Moore
Danville Community College
21
Vertical Shifts
(rigid transformation)
For a positive real number c,
vertical shifts of y = f(x) are:
1. Vertical shift c units upward:
h(x) = y + c = f(x) + c
2. Vertical shift c units downward:
h(x) = y  c = f(x)  c
Horizontal Shifts
(rigid transformation)
For a positive real number c,
horizontal shifts of y = f(x) are:
1. Horizontal shift c units to right:
h(x) = f(x  c) ; x  c = 0, x = c
2. Vertical shift c units to left:
h(x) = f(x + c) ; x + c = 0, x = -c
Reflections in the Axes
Reflections in the coordinate axes of the
graph of y = f(x) are represented as
follows.
1. Reflection in the x-axis: h(x) = f(x)
(symmetric to x-axis)
2. Reflection in the y-axis: h(x) = f(x)
(symmetric to y-axis)
Arithmetic Combinations
Let x be in the common domain of f and g.
1. Sum: (f + g)(x) = f(x) + g(x)
2. Difference: (f  g)(x) = f(x)  g(x)
3. Product:
(f  g) = f(x)g(x)
f
f ( x)
4. Quotient:  ( x) 
, g ( x)  0
g ( x)
g
Composite Functions
The domain of the composite function f(g(x))
is the set of all x in the domain of g such
that g(x) is in the domain of f.
The composition of the function f with the
function g is defined by
(fg)(x) = f(g(x)).
Two step process to find y = f(g(x)):
1. Find h = g(x).
2. Find y = f(h) = f(g(x))
One-to-One Function
For y = f(x) to be a 1-1 function,
each x corresponds to exactly
one y, and each y corresponds to
exactly one x.
A 1-1 function f passes both the
vertical and horizontal line tests.
VERTICAL LINE TEST
for a Function
A set of points in a coordinate
plane is the graph of
y as a function of x
if and only if no vertical line
intersects the graph at more than
one point.
HORIZONTAL LINE TEST
for a 1-1 Function
The function y = f(x) is a
one-to-one (1-1) function if
no horizontal line intersects
the graph of f
at more than one point.
Existence of an
Inverse Function
A function, f, has
an inverse function, g,
if and only if (iff) the
function f is
a one-to-one (1-1) function.
Definition of an
Inverse Function
A function, f, has an inverse
function, g, if and only if
f(g(x)) = x and g(f(x)) = x,
for every x in domain of g
and in the domain of f.
Relationship between Domains
and Ranges of f and g
If the function f has an
inverse function g, then
domain range
f
x
y
g
x
y
Finding the Inverse of a Function
1. Given the function y = f(x).
2. Interchange x and y.
3. Solve the result of Step 2
for y = g(x).
4. If y = g(x) is a function,
then g(x) = f-1(x).
PRECALCULUS I
Mathematical Modeling
•Direct, inverse, joint variations;
Least squares regression
Dr. Claude S. Moore
Danville Community College
34
Direct Variation Statements
1. y varies directly as x.
2. y is directly proportional to x.
3. y = mx for some nonzero constant m.
NOTE: m is the constant of variation or the
constant of proportionality.
Example: If y = 3 when x = 2, find m.
y = mx yields 3 = m(2) or m = 1.5.
Thus, y = 1.5x.
Direct Variation as nth Power
1. y varies directly as the nth power of x.
2. y is directly proportional to the nth
power of x.
3. y = kxn for some nonzero constant k.
NOTE: k is the constant of variation or
constant of proportionality.
Inverse Variation Statements
1. y varies inversely as x.
2. y is inversely proportional to x.
3. y = k / x for some nonzero constant k.
NOTE: k is the constant of variation or the
constant of proportionality.
Example: If y = 3 when x = 2, find k.
y = k / x yields 3 = k / 2 or k = 6.
Thus, y = 6 / x.
Joint Variation Statements
1. z varies jointly as x and y.
2. z is jointly proportional to x and y.
3. y = kxy for some nonzero constant k.
NOTE: k is the constant of variation.
Example: If z = 15 when x = 2 and y = 3,
find k.
y = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5.
Thus, y = 2.5xy.
Least Squares Regression
This method is used to find the
“best fit” straight line
y = ax + b
for a set of points, (x,y),
in the x-y coordinate plane.
Least Squares Regression Line
The “best fit” straight line, y = ax + b, for a
set of points, (x,y), in the x-y coordinate
plane.
a
 xy   x y
2
2
n x   x 
n
1
b
n
 y  a x 
Least Squares Regression Line

a
X
1
2
4
7
Y
3
5
5
13
 xy   x y
2
2
n x   x 
n
X2
1
4
16
21
XY
3
10
20
33
1
b
n
 y  a x 
Solving for a = 0.57 and b = 3,
yields y = 0.57x + 3.