Memory Aid | Mathematics — Secondary 4 (TS)

Find the result of the following division:

​Simplify the following algebraic expression: ||\dfrac{x-2}{x+4} - \dfrac{3}{-3x-12}||

Find the simplified algebraic expression for the following multiplication: ||(7x + 4) (2x^2 -4x +3)||

​​Sometimes several factorization methods are required to factor the same algebraic expression, and that is why it is important to master them all.

Function form (Standard form)
|y = ax + b| where | a = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1}|

General form
|0 = Ax + By + C| where |A, B, C \in \mathbb{Z}|

With the information provided in the graph below, determine the equation of the line in general form.

​|y = a(bx)^2|

where |b=1| unless working with a specific context.

​​Determine the parabola’s equation with the information provided in the table below.​

For second-degree polynomial functions: |a \ne \dfrac{\Delta y}{\Delta x}|

​​|f(x) = a \sqrt{bx}|

where |b=1| if the function is oriented to the right and |b=-1| if the function is oriented to the left.

​Find the equation of the following square root function:

​|y = a(c)^{bx}|

where

|a:| Initial value

|c:| Base (multiplying factor)

|b:| Calculation frequency

​In 2005, a pond had 500 toads. Its toad population decreases by 5% every three years. If this pace continues, in what year will it have approximately 368 toads?

|f(x) = a \left[ b x\right]|

where

|{\mid}a{\mid} = | Vertical distance or height between two steps

|\dfrac{1}{{\mid}b{\mid}} = | Length of a step

To determine the sign of both |a| and |b,| refer to the direction of the open and closed points, and the variation (increasing or decreasing) of the graph:

A grocery store offers a new reward programme with stamps that provide significant discounts on selected items. To determine the number of stamps given to each customer, the grocery store uses the following graph:

Using this graph, determine the possible purchase amounts if a customer receives |48| stamps.

Marie-Claude decides to get back in shape after a vacation by cycling with her group of friends. A coach guides them and decides what speed to maintain. To prepare the group for the next session, the coach gives them the following graph:

​

The training consists of repeating the same trip for |45| minutes. How many minutes, in total, will Marie-Claude have pedalled at a minimum speed of |16| km/h?

Sketch the graph of the inverse of the following function:

The inverse of a function is not always a function itself. In the cases studied here, these functions are inverses of each other:

• Exponential and logarithmic functions

• A linear function with respect to itself

• A second-degree function (with a restricted domain) and a square root function​

​​As an accountant of a large company, you must give a detailed account of the trends in profits over the past year. To help, here is the graph of the last 12 months.​

In order to properly support your argument, you must study this graph closely before preparing your presentation speech.

At a corner store, a group of workers buys 4 coffees and 6 muffins for |$15.06.| The next day, they buy |3| coffees and |5| muffins for a total of |$11.97.| If the following day they want to buy |6| coffees and |4| muffins, how much will it cost?

At a corner store, a group of workers buys |4| coffees and |6| muffins for |$15.06.| The next day, they buy |3| coffees and |5| muffins for a total of |$11.97.| If the following day they buy |6| coffees and |4| muffins, how much will it cost?

At a corner store, a group of workers buys 4 coffees and 6 muffins for |$15{.}06.| The next day, they buy 3 coffees and 5 muffins for a total of |$11{.}97.| If the following day they buy 6 coffees and 4 muffins, how much will it cost?

What is the simplified value of the following radical: ||\sqrt{45}||

To stand out from other contractors, a construction company suggests houses with roofs of different shapes. Among these choices, we have the following form:

The company manager needs the two missing outer measurements of triangle |(\overline{AB},\overline{BC})| to estimate the production costs. Help him find them.

Considering the angle |\theta| as a reference, we have:

|\sin \theta = \dfrac{\text{Measurement of side opposite to}\ \theta}{\text{Measurement of the hypotenuse}}|

|\cos \theta = \dfrac{\text{Measurement of the side adjacent to}\ \theta}{\text{Measurement of the hypotenuse}}|

|\tan \theta = \dfrac{\text{Measurement of the side opposite to} \ \theta}{\text{Measurement of side adjacent to} \ \theta}|

The elevation angle of a house's roof trusses must be a minimum of |25^\circ| for building standards to be met. To ensure that this constraint is respected, a manufacturer decides to establish this angle at |35^\circ.| If the length of the roof truss is |13| metres, what will be the measurements of the other two sides?

​

To determine the route a helicopter should take to pick up people in the woods who need help, a map of the region has been triangulated with the helicopter’s current location, the hospital and the people who are in distress.

According to this drawing, what angle of orientation should the helicopter take to reach the people as quickly as possible?

Find the area of the following triangle:

Due to machinery problems at a construction company, employees have to assemble the triangular-shaped roof trusses themselves to complete the construction on a house. All of the roof trusses must be identical.

Using the information provided above, demonstrate that these two constructions are congruent (isometric).

The city is organizing a family run for a community fundraiser. They want the path taken by adults to be similar to that taken by the children.

Considering the information given above, demonstrate that the two paths are similar.

|\text{Distance} = \sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}|

where

|(x_1, y_1):| Coordinates of the starting point of the segment

|(x_2, y_2):| Coordinates of the ending point of the segment

To determine how much gasoline an airplane must have to complete a Montreal-Paris flight, both cities are represented on a Cartesian plane graduated in kilometres.

What is the distance in kilometres between these two cities?

Consider |(x,y),| the coordinates of the desired division point. ||x=x_1+ \dfrac{a}{b} (x_2-x_1)|| ||y=y_1+ \dfrac{a}{b} (y_2-y_1)||

where

|(x_1,y_1):| Starting point of the segment

|(x_2,y_2):| Endpoint of the segment

|\dfrac{a}{b}:| Fraction that defines the division of the segment (part to whole)

Every morning, you wait at the bus stop for the bus to take you to school. You noticed that for the stop to be on-route for the other students in the area, it divides the segment of the street from your house to the school into a ratio of |1:4.|

Using the information provided, determine the coordinates of your bus stop’s location.

It is important to differentiate between the two types of notations used to illustrate the portion associated with a division point. This is done in order to then use the appropriate notation for the formula: ||\begin{matrix}\text{Ratio}\\ a:b\end{matrix}\Leftrightarrow\begin{matrix}\text{Fraction}\\ \dfrac{a}{a+b}\end{matrix}||

What is the equation of the line that is parallel to the one identified in the Cartesian plane below and passes through point C?

What is the equation of the line that is perpendicular to the one identified in the Cartesian plane below and passes through point |C|?

​​Do not forget the types of events that have been seen in previous years (certain, probable, impossible, elementary, complementary, compatible, and incompatible).

Consider |a,| the odds for and |b,| the odds against. Therefore:

A ratio of odds for |= a : b \Rightarrow \dfrac {a}{a+b}|

RA ratio of odds against |= b : a \Rightarrow \dfrac {b}{b+a}|

Thus, the net gain is obtained according to the following proportion: ||\dfrac{\text{Bet Amount}}{\text{Net Gain}}=\dfrac{\text{No. of chances bet on}}{\text{No. of total chances}}||

During the days of the Quebec racetrack, it was possible to bet on the victories of racehorses. Each horse had a rating which quantified its chances of winning. For the last race, a fan bet |$20| on the victory of a horse whose odds for winning were |1:14.| What was the potential payout (net gain) of his bet?

For some boxing matches, the outcome of a match can be bet upon. Each fighter has a rating which quantifies their chances of winning. The champion has |44:1| chances of winning the next fight. If a fan bet |$10| against the champion's victory, what would be their net gain?

|\mathbb{E} = (p_1 x_1 + p_2 x_2 + \dots + p_i x_i) - M|

where

|p_i =| Probability of occurrence of the event |i|

|x_i =| Amount associated with the event |i|

|M =| Initial bet amount

​​If |\mathbb{E}=0,| then the game is fair.
If |\mathbb{E}<0,| then the game is unfavourable to the player.
If |\mathbb{E}>0,| then the game is favourable to the player.

To finance the school's freestyle ski team, organizers set up a fundraising activity for which it is possible to win the following prizes:

• A weekend family ski pass (value of |$800|)

• Two alpine ski season tickets (value of |$500| each)

• Four pairs of skis (value of |$300| each)

• Eight lift tickets, valid for one day (value of |$45| each)

There are |336| tickets to sell. What should the selling price of a ticket be to ensure the raffle is fair?

|P(A \mid B) = \dfrac{P (A \cap B)}{P (B)}|

where |P(A) >0|

During the previous month, listeners of a Quebec radio channel had the chance to win a trip to Walt Disney World. Before randomly drawing for the winner, the broadcaster provided an overview of the participants:

With the knowledge that he was given the raffle ticket as a gift, what is the probability that the winner will be the father of a family of three children?

​|EM = \dfrac{\sum \mid​ x_i - \overline {x} \mid}{n}|

where

|x_i| each data point
|\sum| the sum
|n| the total number of data

Over the last month, 11 houses in the same neighbourhood were sold for the following prices:

|\color{blue}{$\ 156\ 700},\color{red}{$\ 158\ 900},$\ 159\ 000,$\ 162\ 500,$\ 164\ 100,$\ 167\ 400,$\ 172\ 000,$\ 175\ 000,$\ 178\ 100,$\ 179\ 000,$\ 183\ 000.|

For statistical purposes, calculate the mean deviation of this distribution.

For a sample: |s= \sqrt{\dfrac{\sum(x_{i}-\overline{x})^{2}}{n-1}}|

For a population: |\sigma = \sqrt{\dfrac{\sum(x_i-\mu)^2}{N}}|

where

|\sum| a sum of several elements.

|x_i| the |i^\text{th}| distribution value.

|\overline{x}| the sample mean and |\mu,| the average of the population.

|n| the sample size and |N,| the size of the population.

In some university courses, professors assign marks based on exam scores and the standard deviation of the distribution. What is the standard deviation of the following distribution?

A new company has increased its profits in the last five years and wishes to expand its production centre. However, the owners want to ensure that the economic growth of their company is positive and strongly correlated. To break it all down, here's a count of business income for the past 30 weeks.

In your opinion, is the economic growth of the company positive and strongly correlated?

To take stock of the success of students who enroll in adult education, the administration team is interested in the correlation between absenteeism (in hours) and the students’ final grades (in %). To analyze everything properly, they grouped the data into a scatter plot:

What is the correlation coefficient of this study?

Before a new condo tower is built and the landscaping is done, the heights of the surrounding trees are measured to ensure they do not obscure the view for at least 20 years. The following table of values is used to estimate the height of the view:

Using this information, determine how high the first balconies should be for the view to remain unobstructed by trees.

Although the situation and data are the same, it is normal that the final answer varies depending on the method used (Median-Median method or Mayer method).

Since these methods are used to estimate rather than to predict outcomes with certainty, there may be a difference between the two outcomes.

Before a new condo tower is built and the landscaping is done, the height of the surrounding trees are measured to ensure they do not obscure the view for at least 20 years. The following table of values is used to estimate the height of the view:

Using this information, determine how high the first balconies should be for the view to remain unobstructed by trees.

​Although the situation and data are the same, it is normal that the final answer varies depending on the method used (Median-Median method or Mayer method).

Since these methods are used to estimate rather than to predict outcomes with certainty, there may be a difference between the two outcomes.