Transcript Chapter 11 notes

Section 11 – 1 Simplifying Radicals
• Multiplication Property of Square Roots: For
every number a > 0 and b > 0, ab  a  b
• You can multiply numbers that are both under
the radical and you can separate a number
under a radical into two radicals being
multiplied by each other
• First 10 perfect squares: 1, 4, 9, 16, 25, 36,
49, 64, 81, 100 (know these)
• Simplifying a radical is when you rewrite a
radical so that all of the perfect squares have
been factored out
• The end result is said to be in simplified
radical form
• Simplify each expression
• Ex1. 243
• Ex2. 28x7
• Ex3. 12  32
• Ex4. 7 5 x  3 8 x
• Division Property of Square Roots: For every
number a > 0 and b > 0, a  a
b
b
• You use this property just as you would with
multiplication
• Fractions are not allowed to have radicals in
the denominator
• You must rationalize the denominator
• 1) Simplify the radical(s)
• 2) Multiply the numerator and denominator
by the radical remaining in the denominator
• Simplify
• Ex5.
75 x5
48 x
3
• Ex6.
7
• Ex7.
11
12x 3
Section 11-2 The Pythagorean
Theorem
• The Pythagorean Theorem is applied only to
RIGHT triangles
• You can use this theorem to find the length
of missing sides
• The two shortest sides of a right triangle are
called the legs (they must meet at a 90°
angle)
• The longest side is called the hypotenuse (it
is directly across from the 90° angle)
• The Pythagorean Theorem: a² + b² = c²
where a and b are the legs and c is the
hypotenuse
• Find the length of the missing side (to the
nearest tenth)
• Ex1. a = 8, b = 12, c = ?
• Ex2. a = 20, b = ?, c = 41
• Ex3.
67 ft
25 ft
x
Section 11 – 3 The Distance and
Midpoint Formulas
• The Pythagorean Distance Formula: The
distance d between any two points (x1, y1)
and (x2, y2) is d   x  x    y  y 
• Ex1. Find the distance between A(-3, 7) and
B(5, -4). Show work.
• Ex2. Find the perimeter of ∆XYZ with X(-3, 4)
Y(-1, -5) and Z(2, 2). Show work.
• The Pythagorean Distance formula can be
derived from the Pythagorean Theorem
2
2
1
2
2
1
• If you are asked to give an answer in exact
form, you are to give it in simplified radical
form
• The midpoint of a segment is the point that
divides the segment into two equal segments
• The Midpoint Formula: The midpoint M of a
line segment with endpoints A(x1, y1) and
 x1  x2 y1  y2 
B(x2, y2) is 
,


2
2

• Ex3. If segment CD has endpoints C(-4, 7)
and D(3, -2), find the midpoint of CD. Show
work.
Section 11 – 4 Operations with Radical
Expressions
• Radicals are like radicals if they have the
same radicand (the same number under the
radical symbol)
• Unlike radicals have different numbers under
the radical
• You can add and subtract like radicals, just
as you could with like terms
• Ex1. Simplify
A)
4 5 7 5
B)
3 5  8 20
• You can distribute with radicals as well
(remembering that you can multiply radicals
together and then simplify them if possible)
• Ex2. Simplify 10  6  5
• Ex3. Use FOIL & then simplify  3  2 7  3  4 7 
• Conjugates are the sum and the difference of
the same two terms (i.e. 3  7 and 3  7
are conjugates)
• The product of two conjugates results in the
difference of two squares (an integer)
• To rationalize the denominator of an
expression that has an addition or subtraction
radical expression in the denominator, you
must multiply the numerator and denominator
by the conjugate of the denominator
5
• Ex4. Rationalize the denominator 3  11
• You should never leave a radical in the
denominator of the a fraction!
Section 11 – 5 Solving Radical
Equations
• A radical equation is an equation that has a
variable as a radicand
• Remember that the expression under a
radical must be nonnegative
• Ex1. Solve each equation.
a) a  8  6
b) x  7  9
• If an equation has radical expressions on
both sides, square each side and then solve
• Ex2. Solve 3m  7  5m  13
• When you solve an equation by squaring
each side, you create a new equation. This
new equation may have solutions that do not
solve the original equation. See page 609
• These solutions that do not solve the original
equation are called extraneous solutions
• Ex3. Solve
a) m  m  8
b) 6 x  9  4
• You should make a table of values to create
an accurate graph
Section 11 – 6 Graphing Square
Root Functions
• A square root function is a function that
contains the independent variable in the
radicand
• The parent function for square root functions
is y  x
• The graph of the parent function is the
positive half (because radicands can’t be
negative) of a sideways parabola (see page
614)
• The domain of a function contains all
possible values of the independent variable
• The domain of the parent function is
{x: x > 0}
• You can find the domain by graphing and
looking at the graph or you can determine
algebraically what values can meaningfully
be substituted for x
• Ex1. Find the domain of y  x  2
• The equation y  x  k is a translation of the
parent function by k units up
• The equation y  x  k is a translation of
the parent function by k units down
• The equation y  x  h is a translation of
the parent function by h units to the left
• The equation y  x  h is a translation of
the parent function by h units to the right
• Ex2. Graph each equation
a) y  x  3
b) y  x  5
Section 11 – 7 Trigonometric Ratios
• There are three trigonometric ratios: sine
(sin), cosine (cos), and tangent (tan)
• These ratios describe a specific relationship
between an angle in a RIGHT triangle and
two of the sides of that triangle
sin 
opposite
hypotenuse
cos 
adjacent
hypotenuse
tan 
opposite
adjacent
• SOHCAHTOA should help you remember
these ratios (if you spell it correctly)
• Use a capital letter to represent an angle
• Open to page 621 to see how to identify
adjacent leg vs. opposite leg vs. hypotenuse
• Ex1. Use the triangle below to find
a) sin X
b) cos X
c) tan X
X
5 ft
3 ft
Y
4 ft
Z
• You can use your calculator to find the value
of trigonometric functions
• Make sure your calculator mode is in degrees!
• Ex2. Find the value of each expression.
Round to the nearest thousandth.
a) sin 130°
b) cos 130°
c) tan 130°
• You can use SOHCAHTOA to find the lengths
of missing sides of a right triangle
• Ex3. Find the length of x.
37°
x
29
• An angle of elevation is an angle from the
horizontal up to a line of sight (see page 623)
• An angle of depression is an angle measured
below the horizontal line of sight (see page
624)
• You can use angle of elevation and angle of
depression with trigonometric functions to
solve for missing lengths (see example 4 on
page 623 and example 5 on page 624)