Download Chapter Twelve: Radicals, Functions, and Coordinate Geometry

Chapter Twelve: Radicals, Functions, and Coordinate Geometry
Section One: Operations with Radicals
The square root of a number is the number that we can multiply by itself to give us that
number. For example, the square root of 16 is 4 since 4 times itself gives us 16
16  4 . We discussed in chapter 10 some of the principles of radicals or roots. We


also discussed the meaning of the symbol  .
EX1: Try to find the square root of the following. Estimate if necessary.
a. 121
b.  225
c.  22
d. 45
We can add or subtract radicals if they have the same radicand (the number under the
radical).
EX2: Simplify the radical expressions.
a. 7 3  9 3
b. 6  4 2  9  3 2
c. 4 5  7  2 7  9 5
d. m n  n
We can also simplify roots by splitting them. All roots can be multiplied together and
therefore can also be factored. This is the multiplication property of square roots.
a b  ab which means ab  a b . An example of this is as follows: 2 3  6
or we can break a root apart like this 12  4  3 . We use this method to write square
roots in simplest form.
EX3: Write each in simplest form. (Keep in mind that we are finding principle roots so
use the absolute value bars where necessary).
a. 18
b. 72
c. 40
d. 500
e.
y6
f.
r 6 s3
g.
45x 6 y 3 z 4
EX4: Simplify.

a. 2 3

2
b.
8  14
c.
3


6 5


d. 2  7 2  7

We can split division the same way we do multiplication (ex.
EX5: Simplify.
9
a.
121
5
b.
16
4
c.
7
d.
x3
y2 z4
12
12
).

13
13
Section Two: Square-Root Functions and Radical Equations
A square root function is a function that
contains at least one radical. The simplest of
which is y  x (This function is the inverse of
y  x 2 ). Think about what we can take the
square root of. The domain of this function is
zero and positive numbers. Since we are talking
about the principle root, the range is positive
real numbers. The graph is seen to the right.
One example of a radical function deals with a pendulum. The motion of a pendulum is
l
described by the equation t  2
where t is the time it takes to make one full swing,
g
l is the length in centimeters, and g is the acceleration due to gravity (980 centimeters
per second squared).
EX1: Determine the time in seconds that is takes a 120-centimeter pendulum to make one
complete swing. Determine the length in centimeters of a pendulum that takes 1.5
seconds to make one complete swing.
We can solve radical equations using the fact that squaring is the inverse of square
rooting. We use the steps below to solve radical equations:
1. Isolate the square root if there is only one
2. Separate the square roots if there is more than one
3. Square both sides (FOIL if necessary)
4. Clean up and solve now if possible
5. Start over with step one if necessary
6. Check the solution(s)!
EX2: Solve the equations.
a. x  3  3
b. 2 x  3  4
c. 3x  4  x
d. 4 x  5  x
Remember that we solve some quadratic equations using radicals.
EX3: Solve each equation with radicals.
a. x 2  72
b. x 2  200
c. x 2  82  152
d. x 2  y 2  z 2
EX4: The area of a circular flower bed is 120 square feet. What is the diameter of the
bed? (The area of a circle: A   r 2 )
EX5: Solve the equations by using radicals.
a. x 2  10 x  25  49
b. x 2  12 x  36  1
Section Three: The Pythagorean Theorem
The Pythagorean Theorem is a method used to
find a missing side length of a right triangle when
we know the lengths of the other two sides. The
longest side of a right triangle is called the
hypotenuse. It is always across from the right
angle. The other two sides are called the legs of
the triangle.
The Pythagorean Theorem says that the sum of the squares of the legs is equal to the
square of the hypotenuse, or a 2  b 2  c 2 where a and b are the legs and c is the
hypotenuse.
EX1: Find the unknown lengths.
EX2: An airplane leaves an airport and flies 48 km due north and 20 km due east. At that
point, how far is the airplane from the airport?
EX3: Find the missing part of the right triangles.
a. a  3, b  ?, and c  5
b. a  ?, b  6, and c  25
c. a  0.75, b  ?, and c  1.25
We can also use the theorem to determine if we have a right triangle by looking at the
side lengths. If a 2  b 2  c 2 then the triangle is a right triangle. (c is the longest side)
EX4: Do the following lengths define a right triangle?
a. 5, 12, and 13
b. 6, 7, and 8
c. 6, 12, and 6 3
Section Four: The Distance Formula
We can find the distance between two points on a coordinate plane by using the distance
formula. The distance formula is simply a variation of the Pythagorean Theorem.
d
 x2  x1    y2  y1 
2
2
EX1: Find the distance between each pair of points. Round answers to the nearest
hundredth.
a. 11,7  and  5, 1
b.  3,6 and  2,8
We can prove that three points form a right triangle by using the inverse of the
Pythagorean Theorem along with the distance formula.
EX2: Given vertices  3, 4 ,  2, 2 , and  0,3 , determine whether the points form a
right triangle.
The midpoint formula tells us the point that is directly between two other points in a
coordinate plane.
 x  x y  y2 
midpoint   1 2 , 1

2 
 2
We are simply averaging the x’s and averaging the y’s
EX3: Find the midpoint of the following pair of points.
a. 11,7  and  5, 1
b.  3,6 and  2,8
EX4: The streets of a city are laid out like a coordinate plane, with City Hall at the origin.
Alan lives 5 blocks west and 4 blocks south of City hall, and his friend Mona lives 7
blocks east and 2 blocks north of City Hall. They want to meet midway between their
homes. At what location should they meet?
EX5: The center of a circle is  3, 5 . One endpoint of a diameter is at  2, 3 . What
are the coordinates of the other endpoint of the diameter?
Section Five: Geometric Properties
In this lesson we will discuss several properties of a circle. A circle is the set of all points
that are an equal distance from a given point called the center of the circle. The distance
from the center to any of these points is called the radius. We can use the distance
formula to derive the equation of a circle.
 x  h   y  k 
2
2
 r2
where (h,k) is the center and r is the radius
EX1: Find the equation of the circle with the given center and radius.
a. C  0,0 and r  1
b. C  2,5 and r  5
c. C  2,3 and r  4
We can use the midpoint formula to derive another theorem of geometry called the
triangle midsegment theorem. Triangle Midsegment Theorem: The segment joining the
midpoints of two sides of a triangle is parallel to the third side and is half the length of
the third side.
EX2: Test the Triangle Midsegment Theorem for the triangle with the given coordinates:
 0,0 ,  2,7  , and  6,0 .
We will now discuss the geometry of a parabola. Another definition of a parabola is as
follows: A parabola is the set of all points that are an equal distance from a given point
(the focus) and a given line (the directrix).
EX3: Use the distance formula to write an equation that describes the set of all points that
are an equal distance from the point  4,2  and the x-axis.
Section Six and Seven: The Sine, Cosine, and Tangent Function
Trigonometry is a branch of mathematics that deals mainly with triangles and special
ratios between the sides of the triangles. The most basic trigonometry deals only with
right triangles. Before discussing these ratios we will look at the names for the three sides
of a right triangle. The longest side of a right triangle is the side across from the right
angle. It is called the hypotenuse. The next two sides we define depending on which of
the two acute angles we are referencing. The side that makes up the angle along with the
hypotenuse is called the adjacent side. The side that does not touch the angle is called the
opposite side.
In the figure above, angle C is the right angle.
If referencing angle A:
Side 1 is the opposite side
Side 2 is the adjacent side
Side 3 is the hypotenuse
If referencing and B:
Side 1 is the adjacent side
Side 2 is the opposite side
Side 3 is the hypotenuse
We will discuss three trigonometric functions that describe the relationship between the 3
sides. (We will use the Greek letter theta,  , to name our angle in these formulas.)
Sine of an angle
Cosine of an angle
Tangent of an angle
opp
adj
opp
sin  
cos 
tan  
hyp
hyp
adj
EX1: If opposite is 3, adjacent is 4, and hypotenuse is 5, find the three trig function
values.
EX2: Find the values of the six trigonometric functions of
X and Y in the triangle at the right. Give the exact and
approximate answers rounded to the nearest thousandth.
EX3: Use the calculator to evaluate the following
expressions.
a. sin30
b. cos60
c. tan75
d. sin 1 0.235
e. cos 1 0.168
f. tan 1 1.5
EX4: Use your trig functions to find the missing side length.
a. Use sine if   30 and the hypotenuse is 12 to find the opposite.
b. Use cosine if   48 and the hypotenuse is 23 to find the adjacent.
c. Use tangent if   65 and the adjacent is 35 to find the opposite.
We can use the six trig functions to find missing pieces of our triangle.
EX4: For the triangles below, find the missing side lengths.
a.
b.
We can use these procedures to find an angle of depression or angle of elevation
(inclination).
EX5: The height of an observation tower in a state park is 30 feet. A ranger at the top of
the tower sees a fire along a line of sight that is at a 1° angle of depression. How far is the
fire from the base of the tower? Round your answer to the nearest foot.
When given the sides of a triangle and we are looking for the sides we use the inverse
functions: sin 1 , cos 1 , and tan 1 . We sometimes call these functions arcsine, arccosine,
and arctangent.
EX6: Solve the following triangles (Find all sides and angles). Keep in mind that the
angles of all triangles add up to 180 degrees.
a. fj
b.
Section Eight: Introduction to Matrices
A matrix (plural: matrices) is simply a way of displaying data in a table form. We draw a
matrix in brackets. Matrices are usually named with a capital letter.
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
34 40 23 17 
E

 29 26 30 22
Matrices are divides into rows (horizontal) and columns (vertical). The dimensions tell
how big the matrix is. We measure dimension as row column . Matrix E above is a 2x4
matrix. Therefore it has eight entries or pieces. We name each entry in the following way:
e12  40 (first row, second column) and e24  22 (second row, fourth column). A matrix
with the same number of rows and columns are called square matrices. Two matrices are
equal only if every entry in each are equal.
EX1: Are the following matrices equal?
3
 0.06

64  ? 
23 

   50
1  11 48  


6   110  12 
We can add or subtract matrices simply by adding or subtracting the corresponding
entries. Therefore, we cannot add or subtract unless the dimensions of the matrices are
the same.
3 5
1 4
 0 1
 3 1 


.
EX2: Let X 
and Y  
 4 6 
 2 1




8 1
2 5
a. Find X  Y
b. Find X  Y
There are two types of matrix multiplication. The first type of multiplication is scalar
multiplication. It is simply the same thing as the distributive property.
EX3: Perform the scalar multiplication.
3 0 2 
a. 4 

5 2 1 
 3 0 
 5 2 

b. 1 
 6 4 


 1 1 
The next type of multiplication is when we try to multiply two matrices together.
Dimensions do not have to be the same in order to multiply two matrices. However, not
all matrices can be multiplied. Two matrices can only be multiplied if the first matrix has
the same number or columns as the second matrix has rows.
Can Multiply
 2  4 4  3
Can’t Multiply
 2  43  4
3  55 1
1 7 7  9
1 25 1
 6  2 4  5
The number one rule to remember when multiplying matrices is to multiply rows by
columns and add.
EX4: Multiply the following matrices.
 2 1  1 3
a. 


 4 2  2 4
 3 4  1 2 3
b. 


 2 5   2 1 3 
 6 3
c.  1 0 1  2 0 


 4 5 
3 5
d.  4 1 2 

1 0 
There is an identity element for matrix multiplication. The identity element is a square
matrix with ones on its main diagonal and zeros everywhere else. Below are the 2  2 ,
3  3 , and 4  4 identity matrices.
1 0 0 0 
0 1 0 0 
1 0 


I
I 22  
I

33
44



0
1
0
0
1
0




0 0 0 1 
EX5: State the identity matrix for the matrix and verify by using multiplication.
 3 2 1
 2 3 1


 4 4 5
1 0 0 
  0 1 0 
 0 0 1 