Review for Test 3
Solving Radical Equations
Method of Solving Radical Equations
Step 1: Begin by isolating the radical expression on one
side of the equation. If there is more than one
radical expression, choose one of the radical
expressions to isolate on one side.
Step 2: Raise both sides of the equation by the power
necessary to “undo” the isolated radical. That is,
if the radical is an nth root, raise both sides to
th
the n power.
Solving Radical Equations
Method of Solving Radical Equations (Cont.)
Step 3: If any radical expressions remain, simplify the
equation if possible and then repeat steps 1 and
2 until the result is a polynomial equation.
When a polynomial equation has been
obtained, solve the equation using polynomial
methods.
Step 4: Check your solutions in the original equation.
Any extraneous solutions must be discarded.
Example: Solving Radical Equations
Solve the radical equation.

1  x 1  x
1 x  x 1
1 x

2
  x  1
2
1  x  x2  2x  1
0  x 2  3x
0  x  x  3
x  0, 3
x0
Note that 1  ( 3)  ( 3)  1 , so -3 is
an extraneous solution.
The Cartesian Coordinate System
The Cartesian coordinate
system consists of two
perpendicular real number
lines (each of which is an
axis). The point of
intersection is called the
origin of the system, and
the four quarters defined
by the two lines are called
the quadrants of the plane.
The Distance Formula
Letting  x1 , y1  and  x2 , y2  represent two points
on the Cartesian plane, the distance between
these two points may be found using the
following distance formula, derived from the
Pythagorean Theorem:
d
 x2  x1    y2  y1 
2
2
Example: The Distance Formula
Determine the distance between  1,4  and  3,7  .
d
 x2  x1    y2  y1 
d
 3  1   7  4 
2
d  16  9
d  25
d 5
2
2
2
Recognizing Linear Equations in Two Variables
Linear Equations in Two Variables
A linear equation in two variables, say the variables x
and y, is an equation that can be written in the form
ax  by  c
where a , b, and c are constants and a and b are not
both zero. This form of such an equation is called the
standard form.
Intercepts of the Coordinate Axes
• For an equation in the two variables x and y, it is
natural to call the point where the graph crosses
the x-axis the x-intercept, and the point where it
crosses the y-axis the y-intercept.
• The y-coordinate of the x-intercept is 0, and the x coordinate of the y-intercept is 0.
y-axis
y-intercept
x-intercept
x-axis
Example: Intercepts
Find the x- and y -intercepts of the equation and graph.
3x  4 y  12
3 0   4 y  12
y  3
y-intercept:  0, 3
3x  4  0   12
x4
x-intercept:  4,0 
The Slope of a Line
Let L stand for a given line in the Cartesian plane,
and let  x1 , y1  and  x2 , y2  be the coordinates of
any two distinct points on L. The slope, m , of the
line, L is the ratio
y2  y1
m
x2  x1
which, can be described in words as “change in y
over change in x ” or “rise over run.”
Example: Finding Slope Using Two Points
Determine the slope of the line passing through the
following points.
y2  y1
m
x2  x1
8,1 and  2,3
3 1
m
2  8
2
m
10
1
m
5
Slope-Intercept Form of a Line
If the equation of a non-vertical line in x and y is
solved for y, the result is an equation of the form
y  mx  b.
The constant m is the slope of the line, and the line
crosses the y-axis at b ; that is, the y -intercept of
the line is  0,b . If the variable x does not appear in
the equation, the slope is 0 and the equation is
simply of the form y  b .
Point-Slope Form of a Line
Given an ordered pair  x1 , y1  and a real number m,
an equation for the line passing through the
point  x1 , y1  with slope m is
y  y1  m  x  x1  .
Note that m, x1 , and y1 are all constants, and that x
and y are variables. Note also that since the line,
by definition, has slope m, vertical lines cannot be
described in this form.
The Slopes of Parallel and Perpendicular lines
Important
Parallel lines have the same slope. For example:
1
1
m1  and m2  .
2
2
Perpendicular lines have slopes that are negative
reciprocals of each other. For example:
2
3
m1  and m2   .
3
2
Example: Slopes of Parallel Lines
Find the equation, in slope-intercept form, for the line
which is parallel to the line 8 x  2 y  10 and which
passes through the point  1,5 .
8 x  2 y  10
Step 1: Write equation in
2 y  10  8 x
slope-intercept form.
y  4 x  5
Use slope m  4 to write a new equation that passes
through the point  1,5 .
Step 2: Use point-slope form.
y  5  4  x  1
Step 3: Solve for y to obtain
y  5  4 x  4
slope-intercept form.
y  4 x  1
Solving Linear Inequalities in Two Variables
Step 1: Graph the line that results from
replacing the inequality symbol with .
Solid Line
Dashed Line
 or 
 or 
Non-strict.
Strict.
Points on the
line included in
the solution set.
Points on the line
excluded from the
solution set.
Solving Linear Inequalities in Two Variables
Select a Test Point
Substitute into
the Inequality
true statement
Shade entire half-plane
that includes the test
point
false statement
Shade entire half-plane
that does not include
the test point
Example: Solving Linear Inequalities
Solve the following linear inequality by graphing
its solution set. 3x  2 y  12
x-intercept:  4,0 
y-intercept:  0,6 
Relations, Domains and Ranges
Relations, Domains and Ranges
• A relation is a set of ordered pairs. Any set of
ordered pairs automatically relates the set of first
coordinates to the set of second coordinates, and
these sets have special names.
• The domain of a relation is the set of all the first
coordinates.
• The range of a relation is the set of all second
coordinates.
Functions and the Vertical Line Test
The Vertical Line Test
If a relation can be graphed in the Cartesian plane, the
relation is a function if and only if no vertical line
passes through the graph more than once. If even one
vertical line intersects the graph of the relation two or
more times, the relation fails to be a function.
Implied Domain of a Function
The domain of the function is implied by the
formula used in defining the function. It is
assumed that the domain of the function
consists of all real numbers at which the
function can be evaluated to obtain a real
number: any values for the argument that result
in division by zero or an even root of a negative
number must be excluded from the domain.
Quadratic Functions and Their Graphs
Vertex Form of a Quadratic Function
2
The graph of the function g  x   a  x  h   k where a,
h and k are real numbers and a  0 is a parabola whose
vertex is (h,k). The parabola is narrower than f  x   x 2
if a  1 and is broader than f  x   x 2 if 0  a  1 . The
parabola opens upward if a is positive and downward if
a is negative.
Quadratic Functions and Their Graphs
A quadratic function or a second-degree function of
one variable is any function that can be written in the
form f  x   ax 2  bx  c where a, b, and c are real
numbers and a  0.
The graph of any quadratic function is a roughly Ushaped curve known as a parabola.
Commonly Occurring Functions
Piecewise-Defined Function
A piecewise-defined function is a function defined in
terms of two or more formulas, each valid for its own
unique portion of the real number line. In evaluating a
piecewise-defined function f at a certain value for x, it
is important to correctly identify which formula is valid
for that particular value.
Example: Piecewise-Defined Function
Sketch the graph of the function
2 x  2 if x  1
f  x   2
.
 x if x  1
The function f is a linear function on the interval
 , 1 and a quadratic function on the interval
 1,  . To graph f, we graph each portion
separately, making sure that each formula is applied
only on the appropriate interval. The complete
graph appears to the right, with the points f(–4) = 6
and f(2) = 4 noted in particular. Also note the use of
a closed circle at (–1,0) to emphasize that this point
is part of the graph, and the use of an open circle at
(–1,1) to indicate that this point is not part of the
graph.
Example: Variation Problems
For the following phrases, write the general formula that applies.
a. “y varies inversely as the n ͭ ͪ power of x”
y
k
xn
b. “y is directly proportional to the n ͭ ͪ power of x”
y  kx n
c. “y is inversely proportional to the n ͭ ͪ power of x”
k
y n
x
d. “y varies directly as the n ͭ ͪ power of x”
y  kx n
Shifting, Stretching and Reflecting Graphs
Horizontal Shifting
Let f(x) be a function whose graph is known, and let h
be a fixed real number. If we replace x in the definition
of f by x – h, we obtain a new function g  x   f  x  h .
The graph of g is the same shape as the graph of f, but
shifted to the right by h units if h > 0 and shifted to the
left by h units if h < 0.
Shifting, Stretching and Reflecting Graphs
Vertical Shifting
Let f(x) be a function whose graph is known and let k
be a fixed real number. The graph of the function
g  x   f  x   k is the same shape as the graph of f, but
shifted upward if k > 0 and downward if k < 0.
Shifting, Stretching and Reflecting Graphs
Reflecting With Respect to the Axes
Let f(x) be a function whose graph is known.
1. The graph of the function g(x)= –f(x) is the
reflection of the graph f with respect to the x-axis.
2. The graph of the function g(x) = f(–x) is the
reflection of the graph of f with respect to the
y-axis.
Shifting, Stretching and Reflecting Graphs
Stretching and Compressing
Let f(x) be a function whose graph is known, and let a
be a positive real number.
1. The graph of the function g(x) = a f(x) is stretched
vertically compared to the graph of f if a > 1.
2. The graph of the function g(x) = a f(x) is compressed
vertically compared to the graph of f if 0 < a < 1.
Shifting, Stretching and Reflecting Graphs
Order of Transformations
If a function g has been obtained from a simpler function f
through a number of transformations, g can usually be
understood by looking for the transformations in this order:
1. Horizontal shifts
2. Stretching and compressing
3. Reflections
4. Vertical shifts
Symmetry of Functions and Equations
y-axis Symmetry
The graph of a function f has y-axis
symmetry, or is symmetric with
respect to the y-axis, if f(−x) = f(x) for
all x in the domain of f. Such functions
are called even functions.
Symmetry of Functions and Equations
Origin Symmetry
The graph of a function f has origin
symmetry, or is symmetric with
respect to the origin, if f(−x) = −f(x)
for all x in the domain of f. Such
functions are called odd functions.
Example
• Sketch the graphs of the following relations.
1
a. f  x   2
x
b. g  x   x 3  x
c. x  y 2
• Solutions:
a.
This relation is actually a function, one
that we have already graphed. Note
that it is indeed an even function and
has y-axis symmetry:
f  x 
1
 x
2
1
x2
 f  x .

Example (cont.)
b.
We do not quite have the tools yet to
graph general polynomial functions,
but g(x) = x3 − x can be done. For one
thing, g is odd: g(−x) = −g(x) (as you
should verify). If we now calculate a
3
 1
few values, such as g(0) = 0, g     ,
 2
8
g (1) = 0, and g(2) = 6, and reflect these
through the origin, we begin to get a
good idea of the shape of g.
Combining Functions Arithmetically
Addition, Subtraction, Multiplication and Division of Functions
1.  f  g  x   f  x   g  x 
2.  f  g  x   f  x   g  x 
3.  f  g  x   f  x   g  x 
f  x
 f 
x

, provided that g  x   0
4.    
g  x
g
The domain of each of these new functions consists of the common
elements (or the intersection of elements) of the domains of f and g
individually.
Composing Functions
Composing Functions
Let f and g be two functions. The composition of f and
g, denoted f g , is the function defined by
 f g  x   f  g  x  .
The domain of f g consists of all x in the domain of g
for which g(x) is in turn in the domain of f. The function
f g is read “f composed with g,” or “f of g.”
Example: Composing Functions
Given f(x) = x2 and g(x) = x + 5 , find:
a.  f g  6 
g  6   6  5  11
f
g  6   f  g  6  
 f 11
= 112
= 121
First, we will find g(6) by
replacing x with 6 in g(x).
Next, we know that f composed
with g can also be written
f  g  6  . Since we already
evaluated g(6), we can insert the
answer to get f(11).
Continued on the next slide…
Example: Composing Functions (cont.)
Given f(x) = x2 + 2 and g(x) = x + 5 , find:
Again, we know by definition
b.  f g  x   f  g  x  
that  f g  x   f  g  x  .
 f  x  5
= (x + 5)2 + 2
= x2 + 10x + 25 + 2
= x2 + 10x + 27