## Thursday, December 26, 2013

### IB Maths HL and SL - Graph Transformation of Functions

Horizontal and Vertical Shifts

If $f(x)$ is the original function where $c>0$ then the graph of f(x)+c is shifted up $c$ units,
and the graph of $f(x)-c$ is shifted down $c$ units

A vertical shift means that every point$(x,y)$ on the graph of the original function $f(x)$ is transformed to $(x,y\pm c)$ on the graph of the transformed function $f(x)+c \ or \ f(x)-c$ respectively.

The graph of $f(x+c)$ is shifted left $c$ units

The graph of $f(x-c)$ is shifted right $c$ units

A horizontal shift means that every point$(x,y)$ on the graph of the original function $f(x)$ is transformed to $(x\pm c,y)$ on the graph of the transformed function $f(x-c) \ or \ f(x+c)$ respectively.

Reflections

If $f(x)$ is the original function then

The graph of $-f(x)$ is a reflection in the x-axis.

The graph of $f(-x)$ is a reflection in the y-axis.

Absolute value transformation

$|f(x)|$ : Every part of the graph which is below x-axis is reflected in x-axis.

$f(|x|)$ : For $x \geq 0$ the graph is exactly the same as this of the original function.

For $x <0$ the graph is a reflection of the graph for x≥0 in y-axis.

Stretching and Shrinking

If $f(x)$ is the original function, $c>1$ then

The graph of $cf(x)$ is a vertical stretch by a scale factor of $c$

If $f(x)$ is the original function, $0 then
The graph of $cf(x)$ is a vertical shrink by a scale factor of $c$.

A vertical stretch or shrink means that every point $(x,y)$ on the graph of the original function $f(x)$ is transformed to $(x,cy)$ on the graph of the transformed function $cf(x)$.

If $f(x)$ is the original function, $c>1$ then
The graph of $f(cx)$ is a horizontal shrink by a scale factor of $\frac{1}{c}$.

If $f(x)$ is the original function, $0 then
The graph of $f(cx)$ is a horizontal stretch by a scale factor of $\frac{1}{c}$.

A horizontal stretch or shrink means that every point $(x,y)$ on the graph of the original function $f(x)$ is transformed to $(\frac{x}{c},y)$ on the graph of the transformed function $f(cx)$.

Order of Tranformation
When we perform multiple transformations the order of these transformations may affect the final graph. Therefore we could follow the proposed order (with some exceptions) below to avoid possible wrong final graphs.

1. Horizontal Shifts

2. Stretch / Shrink

3. Reflections

4. Vertical Shifts

## Saturday, July 20, 2013

### MAA MathFest

A Joint Meeting with the Canadian Society for the History & Philosophy of Mathematics (CSHPM)
Hartford, CT

MAA MathFest 2013 will be held at the Connecticut Convention Center and Hartford Marriott Downtown in Hartford, Connecticut. There will be a complimentary Grand Opening Reception on the evening of Wednesday, July 31, and the mathematical sessions will take place from Thursday, August 1 through Saturday, August 3.

- See more at: http://www.maa.org/meetings/mathfest#sthash.OZHlftIY.dpuf

# 5 July 2013. Over 127,000 students worldwide are today receiving their results from the May 2013 IB Diploma Programme examination session.

Jeffrey Beard, Director General of the International Baccalaureate, says: “I would like to congratulate all students on their great achievements. Today’s IB diploma graduates can be confident that they possess the skills needed to excel in an increasingly international world, with students uniquely poised for success both at university and beyond. I wish every individual the very best and look forward to hearing of their accomplishments through our global network of IB alumni.”

## Friday, July 5, 2013

### IB Maths Revision Notes - IB Mathematics HL, SL, Studies Revision Notes by www.IBmaths4u.com

IB Maths Revision Notes - IB Mathematics HL, SL, Studies Revision Notes by www.IBmaths4u.com

Complex Numbers for IB Mathematics HL
http://www.ibmaths4u.com/ComplexNumbers.pdf

Mathematical Induction for IB Mathematics HL
http://www.ibmaths4u.com/Mathematical%20Induction.pdf

Trigonometry for IB Mathematics HL
http://www.ibmaths4u.com/Trigonometry.pdf

## Saturday, April 20, 2013

### Normal distribution

Continuous Probability Distributions, Normal Distribution - IB Maths HL

How can we find the standard deviation of the weight of a population of cats which is found to be normally distributed with mean 2.1 Kg and the 60% of the dogs weigh at least 1.9 Kg.

IB Mathematics HL – Continuous Probability Distribution, Normal Distribution

A normal distribution is a continuous probability distribution for a random variable X. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties.
1. The mean, median, and mode are equal.
2. The normal curve is bell shaped and is symmetric about the mean.
3. The total are under the normal curve is equal to one.
4. The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean.

Approximately 68% of the area under the normal curve is between $\mu - \sigma$ and $\mu + \sigma$
and . Approximately 95% of the area under the normal curve is between $\mu - 2 \sigma$ and $\mu +2 \sigma$. Approximately 99.7% of the area under the normal curve is between $\mu - 3 \sigma$ and $\mu + 3 \sigma$

The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.
$Z\sim N(0, 1 ^2)$

Let the random variable $C$ denotes the weight of the cats, so that

$C\sim N(2.1, \sigma ^2)$

We know that $P(C \geq 1.9)=0.6$

Since we don’t know the standard deviation, we cannot use the inverse normal. Therefore we have to transform the random variable $C$ to that of

$Z\sim N(0,1)$ , using the transformation $Z= \frac{C- \mu}{\sigma}$

we have the following

$P(C \geq 1.9)=0.6 \Rightarrow P(\frac{C- 2.1}{\sigma} \geq \frac{1.9- 2.1}{\sigma})=0.6$

$\Rightarrow P(Z \geq \frac{-0.2}{\sigma})=0.6$

Using GDC Casio fx-9860G SD
$\sigma$:1
$\mu$:0
$\frac{-0.2}{\sigma}=-0.2533471\Rightarrow \sigma=\frac{-0.2}{-0.2533471}=0.789 (3 s.f.)$