Math 10C Pre-IB Portfolio Assignment Type 1 Investigating the Quadratic Function
A quadratic function is one where the highest exponent of the independent variable is 2. The quadratic function can be written in the general form of, where a, b, and c are real numbers. However, the quadratic function can also be written in the standard form of , which is sometimes more preferred, where p and q are the x and y coordinates of the vertex, respectively.

The purpose of this task is to investigate the graph of a quadratic equation, the parabola, when the equation is written in the form. By analyzing p and q we can determine the vertex of the graph. Also, by increasing or decreasing the values of p or q, we can translate the parabola vertically and/or horizontally.

First, if we look at the functions y=x2 ,y=x2+3, y=x2-2 we know that all 3 are in general form. To convert general to standard form you will need to use the process called “completing the square” which goes as following: Ex.

y=ax+bx+c

Now if we convert the three functions mentioned above, in standard form respectively they are y=x020, y=x02+3 and y=x02-2. Now if we were to graph these points, either the standard or general form would work.

y=x2 , y=x020 y=x2+3 , y=x02+3

y=x2-2 , y=x02-2

Other examples of these types of graphs could be anything along the line of. An example of a parabola in the form of y=x2q with either a positive or negative q value could be y=x2+5 and y=x2-4. When we graph the two equations they are as...

...Rahul Chacko
IB Mathematics HL Revision – Step One
Chapter 1.1 – Arithmetic sequences and series; sum of finite arithmetic series; geometric
sequences and series; sum of finite and infinite geometric series. Sigma notation.
Arithmetic Sequences
Definition: An arithmetic sequence is a sequence in which each term differs from the
previous one by the same fixed number:
{un} is arithmetic if and only if u n 1 u n d .
Information Booklet
u n u1 n 1d
Proof/Derivation:
u n 1 u n d
u n u n 1 d
u n 1 u1 dn
u n u1 dn
u n u1 n 1d
Derivations:
u1 u n n 1d
u u1
d n
n 1
u u1
n n
1
d
Information Booklet
Sn
n
2u1 n 1d n u1 u n
2
2
Proof:
Sn = u1 + u2 + u3 + …+ un
= u1 + (u1 + d) + (u1 + 2d) + (u1 + 3d) + …+ (u1 + (n − 1)d)
= un + (un − d) + (un − 2d) + (un + 3d) + …+ (un − (n − 1)d)
2Sn = n(u1 + un)
n
S n u1 u n
2
Derivations
2S n
u1
n
2S
u1 n u n
n
2S n
n
u1 u n
un
Geometric Sequences
Definition: A geometric sequence is a sequence in which each term can be obtained from
the previous one by multiplying by the same non-zero constant.
{un} is geometric if and only if
u n 1
r , n where r is a constant.
un
Information Booklet
u n u1 r n 1
Proof:
u n 1
r
un
u n r u n 1
un
u n 1
u n 1
r
u1 r n
u n u1 r n 1
Derivations:
u
u1 nn 1
r
1
u ...

...Grade 10 Academic Mathematics
Ontario Canada Curriculum MathWiz Practice Exam 1
Instructions: Provide solutions where needed with a final statement Pay attention to degree of accuracy required Check your work when finished
Part A
1. 2. 3.
Place your answers only in the space provided. Answers
Determine the slope of the line 3 x 2 y 8 0 . Determine the equation of the vertical line passing through A (5, 11). Determine the distance between the points X (-1, 5) and Y (4, 17). Determine the midpoint between P(-2, 7) and Q (8, 21). Determine if (1, –1) is on the line 3x – 4y – 7 = 0? State the vertex of y 2 x 3 5 . Determine the y-intercept for the parabola y x 2 3 . Determine the first 3 steps for the quadratic function 2 y 3 x 2 1 . Factor 4 x 2 25 . Determine the roots of x 3 x 2 0 .
2 2
4. 5. 6. 7. 8. 9. 10
Yes No Circle one
11. Determine the value of x if the triangles below are similar.
4 10
2 x
MPM 2D page 1
www.mathwiz.ca
Part B.
1.
Show full solutions.
Solve the system of equations below algebraically and verify your answer . (show a check).
3x 2 y 9 2 x 3 y 19
2.
A retailer is blending together peanuts and cashews to create a mixture. If the peanuts sell for 1.25/kg and cashews for $2.79/kg, how many kg of each should he use to make a 100 kg of a mixture that sells for $1.89/kg ? Set up the equations required...

...
ANALYSIS
Physics has a lot of topics to cover. In the previous experiments, we discussed Forces, Kinematics, and Motions. In this experiment, the focus is all about Friction. Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction like fluid friction which describes the friction between layers of a viscous fluid that are moving relative to each other; dry friction which resists relative lateral motion of two solid surfaces in contact and is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces; lubricated friction which is a case of fluid friction where a fluid separates two solid surfaces; skin friction which is a component of drag, the force resisting the motion of a fluid across the surface of a body; internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation and sliding friction.
When surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into heat. This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to heat whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear,...

...
The case between Beauty and Stylish involves concept of a valid contract, pre-contractual statements, express term and misrepresentation.
A valid contract is established between Beauty and Stylish when an offer is accepted and there is intention for both parties to create legal relations. An offer refers to the expression of willingness of the offerer to be contractually bound by an agreement if his or her offer is properly accepted. It has to be clear and certain in terms. It must also be communicated to the offeree before it is being accepted. In addition, the acceptance has to be unqualified, unconditional and made by a positive act. In the case of Beauty and Stylish, a positive act refers to the signing of the contract. All terms of the offer must be accepted without any changes and cannot be subjected to any condition, taking effect only upon fulfillment of that condition. When Beauty and Stylish enter into the agreement, they must intend to bind and bound legally to each other by their agreement. This is the intention to create legal relations between two parties. In the meanwhile, this contract must possess consideration. A contract must therefore be a two-sided affair, with each side providing or promising to provide something of value in exchange for what the other is to provide.
Every contract, whether oral or written, contain terms. The terms of a contract set out the rights and duties of the parties. Terms are the promises and undertakings given by each...

...of the triangle could be expressed as fractions.
This proves that all the numerators of the row are the same.
To further investigate the numerators, I will examine the relationship between the
row number and the numerator, which is shown in the table below. These are the numerators after having changed the 1s on the outside of the triangle to their fraction forms, thus making all the numerators the same.
Row Number (n) | Numerator in Each Row (N) |
1 | 1 |
2 | 3 |
3 | 6 |
4 | 10 |
5 | 15 |
From this table, I can see that the numerator is the result of the current row number added to the numerator of the previous row. That is, if you’re looking for the numerator of row 4, take that number and add the numerator from row 3, which is 6. 4+6=10, and 10 is the numerator of row 4, as shown in the table above. Knowing this, I predict the numerator of the 6th row to be 21. That is, 15 + 6, because 15 was the previous term, and 6 is the current number of rows.
In an attempt to prove my prediction, I will formulate an expression for calculating N through Implicit form (otherwise known as recursive). Implicit form is a way to define the terms of a sequence based on previous terms. In other words I can find the numerator of a row based on previous numerators.
In this expression, let the numerator be N and...

...Math Exam Notes
Unit 1
The Method of Substitution
-Solving a linear system by substituting for one variable from one equation into the other equation
-To solve a linear system by substitution:
Step 1: Solve one of the equations for one variable in terms of the other variable
Step 2: Substitute the expression from step 1 into the other equation and solve for the remaining variable
Step 3: Substitute back into one of the original equations to find the value of the other variable
Step 4: Check your solution by substituting into both original equations, or into the statements of a word problem
-When given a question in words, begin by defining how variables are assigned
Investigate Equivalent Linear Relations and Equivalent Linear Systems
-Equivalent linear equations: equations that have the same graph
-Equivalent linear systems: pairs of linear equations that have the same point of intersection
-For any linear equation, an equivalent linear equation can be written by multiplying the equation by any real
-Equivalent linear systems have the same solution; the graphs of linear relations in the system have the same point of intersection
-Equivalent linear systems can be written by writing equivalent linear equations for either or both of the equations, or by adding or subtracting the original equations
The Method of Elimination
-Solving a linear system by adding or subtracting to eliminate one of the variables
-To solve a linear system by...

...Math Studies: Units Project
Talha Mirza
1A Gilmartin
Dear noble tribe leaders of T-Island, I have created a brand-new system of units for us to use in our daily lives called the T-Dog system. You will find this system to be quite useful, as it includes six unique units which measure temperature, length, time, weight/mass, volume/displacement, and currency. By utilizing these units, life will be made easier for us. Moreover, if it is ever decided to switch to other well-known systems, it is possible as each unit has a simple conversion to the metric system. Lastly, there are 5 situations attached to this proposal which exemplify the simplicity and convenience of using the T-Dog system.
The first unit in the T-Dog system is “Sweats” ( ), a measurement of temperature.
This measurement was invented by the unique Talha Tribe, who were the first ones to employ the use of this ingenious system.
This measurement is used to calculate and measure heat, specifically the temperature of any given atmosphere, and depict the weather.
This measurement is based on the amount of sweat one produces under a given amount of heat; therefore this unit was given the name “Sweats”.
The symbol used to designate the “Sweats” unit is a small water/sweat droplet. This symbol was created to visually represent the unit of “Sweats”.
The “Sweats” unit, in relation to the metric system would convert by the following:
1 Sweat = 10 ° Celsius
The...

...Jonghyun Choe
March 25 2011
MathIB SL
Internal Assessment – LASCAP’S Fraction
The goal of this task is to consider a set of fractions which are presented in a symmetrical, recurring sequence, and to find a general statement for the pattern.
The presented pattern is:
Row 1
1 1
Row 2
1 32 1
Row 3
1 64 64 1
Row 4
1 107 106 107 1
Row 5
1 1511 159 159 1511 1
Step 1: This pattern is known as Lascap’s Fractions. En(r) will be used to represent the values involved in the pattern. r represents the element number, starting at r=0, and n represents the row number starting at n=1. So for instance, E52=159, the second element on the fifth row. Additionally, N will represent the value of the numerator and D value of the denominator.
To begin with, it is clear that in order to obtain a general statement for the pattern, two different statements will be needed to combine to form one final statement. This means that there will be two different statements, one that...