Transcript Writing Equations of Lines

21 Days
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1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Real Numbers
Exponents and Radicals
Algebraic Expressions
Solving Algebraic Equations
Complex Numbers and Interval Notation
Non-Linear Inequalities
Rectangular Coordinate Systems
Equations of Lines
One Day
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If a and b are real numbers, then exactly one
of the following is true.
a = b,
a > b,
or a < b.
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Absolute Value means distance from 0.
Definition of Absolute Value:
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Use absolute value to write: The distance
between x and 8 is at least 12.
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Distance between two points a and b can be
defined as:
Is the following statement true, or false?
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Round 37.2638 to the following number of
significant digits.
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5 digits:
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2 digits:
pg 16 (#
1,11,17,19,33,35,39,41,49ab,51ab,55,61)
One Day
For what purpose do we use exponents?
They are used to express repeated
multiplication.
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Ex’s:
xy
3
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Simplify the following:
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Recall that:
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Rewrite the following examples:
1.
2.
3.
4.
All possible perfect roots must be simplified
No radicals in the denominator. Rationalize!
Use conjugate pairs to rationalize
denominators.
All indices must be reduced. 3/6 => 1/2
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If √ is there, it implies only the principal
square root.
◦ The principal square root is always positive.
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If x2 = 49 you will need 2 answers, use ±.
pg 29 (# 5,7,8 no
calc,13,20,25,35,49,51,53,63, 67,75,79)
Two Days
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Typically written in decreasing powers of x.
The power of the leading term is the degree
of the polynomial.
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Using FOIL and the Distributive Property to
multiply polynomials.
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Writing a polynomial as a product of its
factors.
Essentially undoing the multiplication.
Purposes of factoring:
◦ Simplifying
◦ Rewriting
◦ Solving
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GCF
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Difference of Squares
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Perfect Square Trinomials
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Sum and Difference of Cubes
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Trinomials
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Grouping
pg 44 (#
13,19,21,25,28,29,31,33,34,39,41,45,
47,51,53,57,60-62)
Three Days
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Equation in x – A statement of equality
involving one variable, x.
A number that results in a true statement
when substituted for x is a Solution or Root.
A number Satisfies and equation if it is a
solution to the equation.
To Solve an equation is to find all solutions of
the equation.
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The Domain is the set of all permissible
values that can be assigned for variable(s).
A Domain Restriction is the exclusion of all
values which make the denominator = 0, or
make √ negative.
Ex: What is the domain of:
FACTOR, FACTOR, FACTOR!!!
Bad Example
Good Example
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Another Example
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Solve 3|4x – 1| -5 = 10
pg 60 (# 3,5,7,11,17,19,21,25,27,31,39)
The temperature T (in oC) at which water boils is
related to the elevation h (in meters above sea
level) by the formula
H=1000(100-T)+580(100-T)2 for 95<T<100
a) At what elevation does water boil at a
temperature of 98oC
H=1000(100-T)+580(100-T)2 for 95<T<100
b) The elevation of Mt. Everest is approximately
8840 meters. Estimate the temperature at which
water boils at the top of this mountain. (hint,
use the quadratic formula with x=100-T)
A closed right circular cylindrical oil drum of
height 4 feet is to be constructed so that the
total surface area is 10π ft2. Find the
diameter of the drum.
pg 61 (#
45,46,49,56,57,59,60,63,64,66,68,69)
One Day
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The Imaginary Unit (i) has the following
properties.
Imaginary Numbers are of the form a + bi
where b ≠ 0.
Complex Numbers are of the form a + bi
where a and b are Real Numbers.
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We can add and subtract imaginary numbers
similar to how we add and subtract terms
with variables. Think “like terms.”
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Similarly, we can multiply imaginary numbers
following the same exponent rules we use for
variables.
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Larger powers of i can be simplified by
dividing the power by 4 and using the
remainder to determine the appropriate
value.
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If z = a + bi is an imaginary number, the its
conjugate is z = a – bi.
Complex Conjugates can be used to eliminate
imaginary numbers from the denominators of
fractions. This is very similar to how we
rationalize denominators.
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Eliminate the Imaginary numbers from the
denominator in the following example.
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In this course we will use interval notation to
write infinite solution sets.
pg 73 (# 1,3,5,11,13,19,21,25,29,31,33,39,47,
49,51) Read pg 73-74
Three Days
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When solving inequalities, we treat them just
like linear equations except when we multiply
or divide by a negative we must …..
G
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When solving Non-Linear Inequalities we
must use a number line and evaluate test
points to determine the solution to the
inequality. We will see this in an example on
the next slide.
Common Mistake!!
(x+2)(x-3) < 0 and x+2<0 x-3<0 are
NOT equivalent statements! DO NOT do this!!!
(x+2)(x-3) < 0
 If this were an equation we would have
solutions at x = -2 and x =3.
 We will use these point as critical points on
the number line. We must find where the
product of the factors is (in this case)
negative.
(x+2)
(x−3)
(x+2)(x-3) < 0
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pg 84 (# 6,7,9,20,22,27,29,30) Read 1.6
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Rational Inequalities are non-linear, therefore
we must use a number line, test points, and
resulting sign from the product of factors to
determine our solution.
Bad Example (why?)
Good Example
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pg 84 (# 11,31,36,37,43,45,49,53,54)
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Complete WS and pg 85 (# 42,60,61) Read
1.7
Three Days
d (a, b)  ( x2  x1 ) 2  ( y2  y1 ) 2
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Find the distance between (2,1) and (-4,3)
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Find the midpoint between (1,5) and (-3,8)
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A solution of an equation in x and y is an
ordered pair (a,b) that yields a true statement
if x=a and y=b.
How many solutions exist for the line
y = 2x-1?
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Create and x-y table, plot points, and sketch
the graph of y = -3x + 2.
x
-2
-1
0
1
2
y
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Create and x-y table, plot points, and sketch
the graph of
x
-3
-2
-1
0
1
2
3
y
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The graph in the previous slide is a parabola.
The vertex of a parabola is the min (or max)
of the parabola.
A vertical line through the vertex represents
the axis of symmetry.
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We can determine the intercepts of an
equation using the following method.
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Find all intercepts for
Type of Symmetry
Test for Symmetry
Substitution of –x with x
Symmetry with
leads to the same
respect to the y-axis equation.
Substitution of –y with y
Symmetry with
leads to the same
respect to the x-axis equation.
Substitution of –x with x
Symmetry with
and –y with y leads to
respect to the origin the same equation.
Sample Illustration
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Determine the type of symmetry (if it exists)
for the following equations:
pg 105 (#
2,3,9,14,19,23,27,31,35,36,39,42,47, 51,55)
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Determine the equation of the circle with
radius r=4 and the center at (1,-2).
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Determine the equation of the circle with
radius r=7 and the center at (-3, 2).
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We can find the equations of the upper,
lower, left, and right halves of a circle using
the following method:
Upper and lower halves
◦ Solve for y in terms of x, you will have a ±.
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Right and left halves
◦ Solve for x in terms of y, you will again have a ±.
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Find the equations of the upper, lower, right,
and left semicircles of:
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Find the equations of the upper, lower, right,
and left semicircles of:
pg 105 (# 59,61,63,65,69,77,79) and
pg 784 (# 1,5,9,15,17)
Two Days
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Slope Formula
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Slope Intercept
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Point Slope Form
Horizontal lines are
parallel to the x-axis.
y=b
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Vertical lines are
parallel to the y-axis.
x=a
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Two non-vertical lines are parallel if and only
if they have the same slopes.
m1 = m2
Two lines with slopes m1 and m2 are
perpendicular if and only if
m1m2 = -1
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The Standard Equation of a Line is given by
ax + by = c where a and b ≠ 0, and a,b,c are
integers.
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Write the equation of the line with m=2
passing through (1,3).
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Write the equation of the line that passes
through (4,11) and (-3,3).
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Write the equation of the line that is the
perpendicular bisector of the segment with
endpoints (3,-1) and (-2,6).
pg 123 (# 1,9,17,19,22,24,25,30,32,33,43ab,
47,48)