Week 1

During the first week of Honors Math Analysis, Miss Van Spronsen began our study in Inequalities, Equations and Graphs. It was a bit of review from last year but we still learned new material. There are five lessons in this chapter.

On the first day we began with The Real Line. 

All numbers begin with real numbers other than imaginary numbers. From real numbers, there are rational and irrational numbers. Rational numbers include fractions and decimals while irrational numbers include pi or radical numbers. From Rational numbers there are nonintergers and intergers, which include negative, zero and positive numbers. 

We also reviewed on inequalities and how to graph them.

However, we learned a new way to graph them using set notation or interval notation.

Set Notation

( , )⟶ ◦, <, >, ±∞

[ , ]⟶ ●, ≤, ≥

Another thing we learned on day one was the sign chart method.

The rules are:

1. make the inequality with a zero on one side

2. factor

3. mark where factors are zero

4. in each interval determine the sign

On Day 2, we talked about Absolute Value.

With absolute value, there were methods needed to keep it absolute value, which included:

| x | = { x, if x ≥ 0 } and { -x, if x < 0 }

There are also properties of absolute value which include:

1. | a | = | -a |

2. | a | = 0 if and only if a =0

3. | ab | = | a | | b |

4. | a/b | = | a |/| b |

5. | a+b | ≤ | a |+| b |

We reveiwed on absolute value to solve problems and we also reviewed on how to do it with inequalities.

1. | x | < a if and only if -a<x<a

2. | x | > a iff x>a or x<-a

In an inequality problem, an absolute value problem cannot be ≤ to 0 because an absolute problem would be always positive so it must be = to 0 instead. 

We also looked at the distance between 2 numbers. To finding the distance: d(a,b)= | b-a |

To find the midpoint: m= a+b/2

On Day 3, we looked at The Rectangular Coordinate System. 

This lesson was mostly review on how to graph and shade. We reviewed to sketch points. 

The distance formula we looked at is:

d=[RAD] (x2-x1)^2+(y2-y1)^2

the Midpoint formula is

m= (x1+x2/2 , y1+y2/2)

To find if the vertices listed are a right triangle, you must use the distance formula to find the distance between each point and then to use the pythagorean thereom to see if the vertices are of a right triangle.

On Day 4, we learned and reviewed on Circles and Graphs. 

The equation of a circle is (x-h)^2 + (y-k)^2 = r^2

the center of the circle is (h, k) while the radius is r. 

However some equations were a bit more confusing and completing the square was neccessary. You would split the equation into two and solve for the square using x and y to find the center and the radius. 

We also learned about semicircles:

a circle’s equation is x^2+y^2=r^2

half of that to make a semicirlce is y=±[RAD] r^2-x^2

We then looked at intercepts. To find the intercepts for both x and y, you would plug in a 0 for the other variable in two different problems.

We then looked at the tests for symmetry:

the graph of an equation is symmetric with respect to:

1. the y-axis if replacing x by -x results in an equivalent equation

2. the x-axis if replacing y by -y in an equivalent equation

3. the origin if replacing x and y by -x and -y results in an equivalent equation. 

That is what we learned during the first week of Honors Math Analysis. 

See you Next Week!