## Tuesday, October 30, 2012

When numbers get very large or very small, they become difficult to work with and read. Examples of such

numbers are 0.0000000000043425 and 546,245,000,000,000,000,000. Oftentimes these types of numbers appear in science, such as the distance planets are from Earth or the mass of atoms. Scientific notation is a method to make using such numbers easier by writing them in a simpler form. A positive number in the form N times 10^x , where 1 ≤ N < 10 and x is an integer, is said to be in scientific notation.

When a number is in scientific notation, the decimal point is always after the first non zero number. Some

examples of numbers in scientific notation are

3.5 X 10^3, 5.89 X 10^(-4) and 2.9123 X 10^6

To change a number into scientific notation, we move the decimal point between the first two non zero

numbers. Then we count the number and direction that the decimal point must move to get the original number.

The number of decimal places moved will be the exponent. If we move the decimal point to the right, the

exponent is positive. If we move the decimal point to the left, the exponent is negative.

Examples: Write the following numbers in scientific notation.

1. 567,325,000,000,000,000

First place the decimal point between the 5 and 6. Count how many decimal places we have to move to

the right to get to the end of the number. Notice we have to move 17 decimal places. Therefore,

567,325,000,000,000,000 written in scientific notation is 5.67325 X 10^17.

•Note that the zeros after the 5 are not written when changing to scientific notation. When all the rest of

the numbers are zero they are not written.

2. 0.0000000007982

First place the decimal point between the 7 and 9. Count how many decimal places we have to move to

the left to get back to the beginning of the number. Notice we have to move 10 decimal places.

Therefore 0.0000000007982 written in scientific notation is 7.982 X 10^(-10) .

Be sure to be careful with numbers such as 65.8 X 10^3 and 254.69 X 10^(-4). At first glance, these appear to be in scientific notation, but the decimal point is not after the first non zero number.

To write in scientific notation, multiply the problem out and then convert to scientific notation.

65.8 X 10^3 = 65.8 X 1000 = 65,800

Another way to simplify the above is to move the decimal point 3 places to the right since the exponent is 3.

Now we can change 65,800 into scientific notation, which is 6.58 X 10^4.

In the example 254.69 X 10^(-4), multiply to get 254.69 X 0.0001 = 0.025469.

Another way to simplify the above is to move the decimal point 4 places to the left since the exponent is -4.

Now we can change 0.025469 into scientific notation, which is 2.5469 X 10^(-2) .

In some cases, using scientific notation makes multiplying and dividing very large or very small numbers easier.

## Saturday, October 27, 2012

Suppose an equation is as follows:

y = (x - 1)(x + 5)

First we can find the x intercepts by setting x - 1 and x + 5 equal to 0 and solve for x. When doing so, we get the intercepts to be (1, 0) and (-5, 0). The x coordinate of the vertex is halfway between 1 and -5. Therefore the x coordinate of the vertex is -2. The y coordinate of the vertex is found by substituting -2 for x in the equation.

y = (-2 - 1)(-2 + 5) = -3(3) = -9

Therefore the vertex of the parabola is (-2, -9). The axis of symmetry is the line through the x coordinate of the vertex, or x = -2 in this case.

Suppose the equation is

y = x^2 + 4x - 5.

If the equation is in the form y = ax^2 + bx + c, the x coordinate of the vertex is -b/2a.

a = 1, b = 4. Therefore the x coordinate of the vertex is -4/2(1) = -2.

The y coordinate of the vertex is y = (-2)^2 + 4(-2) - 5 = 4 - 8 - 5 = -9.

The vertex is at (-2, -9) and the axis of symmetry is x = -2.

Suppose the equation is

y + 9 = (x + 2)^2

If the equation is in the form y - k = (x - h)^2, the vertex is (h, k)

Therefore the vertex is (-2, -9) and the axis of symmetry is x = -2.

The equations y = (x -1)(x + 5), y = x^2 + 4x - 5 and y + 9 = (x + 2)^2 are the same, just written in different forms.

## Thursday, October 25, 2012

When thinking of reflexive, think of the word "reflection". When you see your reflection, you see yourself. So reflexive would be something along the lines of "a = a" or "b = b".

When thinking of symmetric, think of the word "symmetry". If you cut something in half and have two equal parts, the object is symmetrical about the cut.

In geometry this is like saying "a = b, then b = a".

The transitive property involves 3 things, notice "trans" meaning three, is part of transitive.

In geometry this is like saying "if a = b and b = c, then a = c"

## Wednesday, October 24, 2012

## Tuesday, October 23, 2012

## Monday, October 22, 2012

Finding the inverse of a function and its graph.

Suppose we have a function f(x) and know that point (c, d) lies on its graph. If f(x) has an inverse, f -1(x), the point (d, c) lies on its graph. The points (c, d) and (d, c) are equidistant from the line y = x. Such points are called mirror images of each other. Notice the graph and the points with respect to the line y = x.

With respect to the line y = x, the points on the graphs of f(x) and f -1(x) are mirror images of each other, so it makes sense to draw the conclusion that the graphs of f(x) and f -1(x) are also mirror images of each other with respect to y = x

.

Examples: a. Find the inverse of f(x) = 2x + 4 and graph f(x) and f -1(x) on one rectangular coordinate system.

Before we find the inverse of f(x), we must determine if an inverse exists. If the function is one-to-one, then it has an inverse. Since f(x) is linear, then it does have an inverse. Recall, to find the inverse we interchange y and x and then solve the equation for y.

f(x) = 2x + 4

y = 2x + 4

x = 2y + 4

x – 4 = 2y

(x – 4)/2 = y

f -1(x) = (x – 4)/2

To graph f(x) = 2x + 4, we know the y- intercept is (0, 4). To find the x- intercept, rewrite the function as the equation y = 2x + 4 and substitute 0 for y. Then solve for x.

y = 2x + 4

0 = 2x + 4

-4 = 2x

-2 = x. Therefore the x- intercept is (-2, 0).

To graph f -1(x) = (1/2)x – 2, we know the y- intercept is (0, -2). To find the x- intercept, rewrite the inverse function as the equation y = (1/2)x – 2 and substitute 0 for y. Then solve for x.

y = (1/2)x – 2

0 = (1/2)x – 2

2 = (1/2)x, 4 = x. Therefore the x- intercept is (4, 0)

Notice the graph of f(x) and its inverse.

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