Producto Integrador Global Matematicas 3 UANL

...

Now I'll plot these points:

[pic 11]

And now I can connect the dots:

Irrational Functions

_________

\/ x+8=x+2

The general method to solve an equation is to replace, in succession, the equation by an equivalent equation.

We tend to square both sides of the equation, but the next equivalence is false.

_________

\/ x + 8 = x + 2 x + 8 = (x + 2)2

Indeed, -4 is solution of

x + 8 = (x + 2)2

but it is not a solution of

_________

\/ x + 8 = x + 2

It is obvious that the following expression is correct.

_________

\/ x + 8 = x + 2 => x + 8 = (x + 2)2

All solutions of the equation on the left side are solutions of the equation on the right side but not vice versa. The equation on the right can have more solutions.

Nevertheless, we will solve irrational equations by squaring both sides. But we must be aware that, by squaring, the new equation can have more solutions than the original one.

At the end, the 'false solutions' are deleted.

There are different ways to find these 'false solutions'. One way is to build suitable inequalities in advance, to find the 'false solutions'. We don't follow this method here.

The simplest way is to test each solution to the given equation. If the given equation is not satisfied for that solution, it is labeled as a 'false solution'.

Let's use this procedure to the equation

_________

\/ x + 8 = x + 2

=> x + 8 = (x + 2)2

...

x = 1 of x = -4

We test these two values to the given equation. We see that -4 is a false solution and it must be deleted. The only solution is 1.

Functions of Variation

Direct Variation

The statement " y varies directly as x ," means that when x increases, y increases by the same factor. In other words, y and x always have the same ratio: = k

where k is the constant of variation.

We can also express the relationship between x and y as:

y = kx

where k is the constant of variation.

Example : If y varies directly as x , and x = 12 when y = 9 , what is the equation that describes this direct variation?

k = [pic 12] = [pic 13] y = [pic 14] x

Inverse Variation

In an inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases.

The number of hours, h, it takes for a block of ice to melt varies inversely as the temperature, t. If it

takes 2 hours for a square inch of ice to melt at 65º,

find the constant of proportionality.

Start with the formula: [pic 15]

Substitute the values : [pic 16]

then solve for k: [pic 17]

Logarithm Function

- Graph y = log3(x) + 2.

This is the basic log graph, but it's been shifted upward by two units. To find plot points for this graph, I will plug in useful values of x (being powers of 3, because of the base of the log) and then I'll simplify for the corresponding values of y.

30 = 1, so log3(1) = 0, and log3(1) + 2 = 2

31 = 3, so log3(3) = 1, and log3(3) + 2 = 3

32 = 9, so log3(9) = 2, and log3(9) + 2 = 4

33 = 27, so log3(27) = 3, and log3(27) + 2 = 5

Moving in the other direction (to get some y-values for x between 0 and 1):

3–1 = 1/3, so log3( 1/3 ) = –1, and log3( 1/3 ) + 2 = 1

3–2 = 1/9, so log3( 1/9 ) = –2, and log3( 1/9 ) + 2 = 0

3–3 = 1/27, so log3( 1/27 ) = –3, and log3( 1/27 ) + 2 = –1

These are the only "neat" points that I'm going to bother finding for my graph. If I feel a need for additional plot points, especially between any two of the points I found above, I can evaluate the function "ln(x) / ln(3)" in my calculator.

The graph of y = log3(x) + 2looks like this:

[pic 18]

Exponential Functions in different contexts

[pic 19]We can use a formula for exponential growth to model the population of a bacteria. Let's say the bacteria population is defined by

where B is the total population and t represents time in hours. While that may look complicated, it really tells us that the bacteria grows by 12 percent every hour. Every time another hour goes by, t goes up by 1, so we have to multiply the population times 1.12 again. The 100 simply sets the initial population at time t=0.

So, how much bacteria remains after 4 hours?

[pic 20]What do we know? We have the formula and the fact that t=4.

Replace t with 4 hours in the formula above and simplify.

[pic 21]

The