Ch+7+0910

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The video should include all 6 members of this wiki page and one concept should be taught or demonstrated. media type="file" key="math wiki.mov" Vocabulary-Katie Conner Real-Life Applications/Problem: Matthew Aguilar Editor: Cheyenne Bohlen Notes 7.1-7.3-Haley Brandt Notes 7.4-7.6- Austin Gibbons Example Problems- Cody Winsor __ 7.1 nth Roots and Rational exponents __ An nth root is the same concept as a square root with different numbers. Ex: 3√8=2 The mathematic formula of an nth root is n√a=b. In this example 3 would be n, 8 would be a, and 2 would be b.

__ Real nth Roots __ n is a positive integer and a is a real number · If n is odd, then a has one real nth root. n√a=a1/n · If n is even and a >0, then a has two real nth roots. ±n√a=±a1/n · If n is even and a=0, then a has one nth root. n√0=01/n=0 · If n is even and a<0, then a has no real nth roots.

Ex:

__ Rational Exponents __ Let a1/n be an nth root of a and let m be a positive integer. · am/n=(a1/n)m=(n√a)m · a-m/n=1/(am/n)=1/(a1/n)m, a≠0

Ex:

__ 7.2 Properties of Rational Exponents __ a and b are both real numbers and m and n are rational numbers.

Real Life Application-
 * Product of Powers Property || am•an=am+n ||
 * Power of a Power Property || (am)n=amn ||
 * Power of a Product Property || (ab)m=ambm ||
 * Negative Exponent Property || a-m=1/am ||
 * Quotient of Powers Property || am/an=am-n ||
 * Power of a Quotient Property || (a/b)m=am/bm ||

The equation f = 440 x 2^(n/12) can be used to find the frequencies of the musical range of an instrument, suchy as a trumpet or guitar. "n" is based on A-440 which is the middle key on a piano, and the "A" key repeats every 12 notes. This is what gives the "n/12" in the equation.

__7.3 Power Functions and Function Operations __ Let f and g be two functions. The new function, h, can be made by combining f and g in any of four ways, addition, subtraction, division, and multiplication.


 * Addition || h(x)=f(x)+g(x) ||
 * Subtraction || h(x)=f(x)-g(x) ||
 * Multiplication || h(x)=f(x)∙g(x) ||
 * Division || h(x)=f(x)/g(x) ||

__ Composition of Two Functions __ The compostion of the function f with the function g is h(x)=f(g(x)). The domain of h is the set of all x-values such that x is in the domain of g and g(x)is in the domain of f.

Real Life Application-

By measuring the size of a fossilized footprint of a dinosaur, scientists can methematically determine the approximate height of the dinosaur that left the print based off of the the ratios that are known because of fossilized bones. These ratios are most often expotential.

**__ 7.4 Inverse Functions __** Functions //f// and //g// are inverses of each other provided: //f//(//g//(//x//)) = //x// and //g//(//f//(//x//)) = //x// The function //g// is denoted by //f// to the -1 power, read as " //f// inverse."


 * __Horizontal Line Test__**

If no horizontal line intersects the graph of a function //f// more than once, then the inverse of //f// is itself a function.

Real Life Application-

One can use an equation such as h= 0.9 x (200-a) to determine a boweling handicap to level the ability range between differing players. "h" is the handicap while "a" is a player's average score. If their average happens to be over 200, their average would be 0.

Per Mrs. Keener
__ ** 7.5 Graphing Square Root and Cube Root Functions ** __

The graph of y = a times the square root of x, starts at the orgin and passes through the point (1,a). The graph of y = a times the cube root of x, passes through the orgin and the points (-1,-a) and (1,a).

__ ** Graphs of Radical Functions ** __

To graph y = a times the square root of x-h, then plus k, or y = a times the cube root of x-h, then plus k, follow these steps.

STEP 1- Sketch the graph of y = a times the square root of x or y = a times the cube root of x. STEP 2- Shift the graph //h// units horizontally and //k// units vertically.

Real Life Application-

If the horsepower within a drag race car is know, then one can determine the speed it will be expected to go in a certain distance. s= 14.8 x p^(1/3) is an example model of this where "s" is the top speed while "p" is the horsepower of the car.


 * __ 7.6 Solving Radical Equations __**

In order to solve a radical equation you must eliminate the radicals or rational exponents and be left with a polynomial equation. The key step is to raise each side of the equation to the same power.

Powers property of equality- If a = b, then a to the nth power = b to the nth power.

Isolate the radical expression on one side of the equation, then solve the new equation using standard procedures.

Real Life Application-

The Beaufort Wind Scale consists of an equation B= 1.69x (s + 4.45)^0.5 -3.49 to determine the approximate force and effect of the wind based on its speed. "B" is the Beaufort Number, which ranges from 0 to 12 to determine the description of the weather. (0 = calm where smoke rises vertically, 12= hurricane with destruction) "s" is the speed of the wind in miles per hour.

__ Vocabulary for Chapter Seven __ __Nth root of a__- If b^n=a, then b is the nth root of a. __Index__ - -n is the index number __Simplest Form__ -to get a radical in this form you must apply the properties of radicals and remove any perfect nth powers (other than 1) and rationalize and denominators. __Like Radicals__ -expressions that have the same index and radicand. __Power Function__ -common type of function which has the form y=ax^b where a is a real number and b is a rational number. ** Section 7.4 ** __Inverse Relation__ -maps the output values back to their original input values. __Radical Functions__ -A function that contains a radical. **Section 7.6** __Radical Equation__ __Extraneous Solution__ -a false solution.
 * Section 7.1 **
 * Section 7.2 **
 * Section 7.3 **
 * Section 7.5 **
 * -**an equation that contains radicals or rational exponents.

__Statistics__ -numerical values used to summarize and compare sets of data. __Measures of Central Tendency__ -three commonly used statistics: mean, median, and mode. __Mean__ - the average of n numbers is the sum of the numbers divided by n. __Median__ -the median of n numbers is the middle number when the numbers are written in order. __Mode__ -the mode of n numbers is the number or numbers that occur most frequently. __Measures of Dispersion__ -statistic that shows how spread out the data are. __Range__ -the difference between the greatest and least data values. __Standard Deviation__ -describes the typical difference between the mean and a data value. __Box-and-Whisker Plot -one type of statistical graph. A “box” encloses the middle half of the data set and the “whiskers” extend to the minimum and maximum data values. Lower Quartile__ -is the median of the lower half. __Upper Quartile__ -is the median of the upper half. __Histogram__ -Type of bar graph where data are grouped into intervals of equal width. __Frequency__ -the number of data values in each interval. __Frequency Distribution__ -shows the frequency of each interval.
 * Section 7.7**