Pre+calculus

Unit 1 Lesson 1 9/7/11 //What are the similarities and differences between a natural number, whole number, and integers?// Natural numbers, whole numbers and integers all can be written without using a fraction or decimal but natural numbers are only positive, whole numbers include zero and natural numbers while integers include whole numbers and negative numbers.

//What is the difference between a rational and irrational number?// //Rational numbers can be represented by repeating or terminating decimal while irrational numbers cannot be represented this way.//

// Explain if the reciprocal of a positive real number must be less then one. If this statement is false prove your argument with an example and explanation. // //The reciprocal of a positive real number can be greater than one if the number, x greater than 0 but less than 1, 0<x<1. Since a number is x/1 the reciprocal of a number is 1/x. For example the reciprocal of .1 or 1/10 the reciprocal of .1 is 10 or 10/1.//

// True or False: An integer is a rational number. Explain your answer and use an example if necessary. // True, all integers can be expressed in a terminating decimal.

// True or False: A rational number is an integer. Explain your answer and use an example if necessary. // // False, a rational number can be an integer but all numbers that can be represented as a terminating or repeating decimal are rational. //

// True or False: A number is either rational or irrational, but not both. Explain your answer and use an example if necessary. // // True, since what separates rational and irrational numbers is whether a number can be represented as a repeating or terminating decimal and a decimal cannot be terminating but not terminating at the same time nor can it be repeating and not repeating at the same. //

//Give an example of a real number set that includes the following elements:// 1.5, 3/2
 * // A rational number that is terminating (represented in both fraction and decimal form) //

1/3, .333333...
 * // A rational number that is infinitely repeating (represented in both fraction and decimal form)//

1: 1 is a natural number because it is a positive number that doesn't have to be represented as a fraction or decimal, it is a whole number because whole numbers include natural numbers and 0, it is an integer because integers include negative and positive numbers that don't have to be represented as decimals or fractions, and it is a rational number because 1 can be represented as a terminating decimal.
 * // A real number that fits at least 4 categories of the real number system and explain verbally how that number fits in each category //

Unit 1 Lesson 3 9/8/11

//What are three methods you can use to find the distance between two points on the coordinate plane? Explain when it is most convenient to use each method.// You can can count the units which is best for horizontal and vertical lines, use the Pythagorean Theorem which is best to use for diagonal lines when you know both coordinates and using the distance formula is best to use in the same situation.

//What are three methods you can use to find the midpoint of two points on a coordinate plane? Explain when it is most convenient to use each method.// You can can count the units of a line segment and divide it in half which is most convenient to use for horizontal and vertical lines, the midpoint formula is useful for diagonal lines and you can also use

//Given the link to the following example, explain in your own words what is going on during each step of the problem.// //In this equation the distance is 5 x1 is 4 x2 is 0 and y1 is -4 while y2 is unknown. 0 - 4 squared is 16. When use foil on y1 and y2 you end up with y squared +8y +32 and on the other side of the equation 5 squared is 25. Combine terms and you end up with 0=y squared + 8y + 7 which can be factored into (y+1)(y+7) The solution is(0,-1) and (0,-7).//

//Unit 1 Lesson 5// //9/11/10//

//What is the standard form equation of a circle with a radius of (0, 0)// x squared + y squared= r squared

//Explain in words how you can find the center of a circle if you are given the two endpoints of the diameter.// Use these 2 coordinate points in the midpoint formula.

//Explain in your own words how you can find the radius of a circle if you are given the center and a point on the circle.// Use those 2 points in the distance formula.

//Using another resource, write the mathematical definition of the word tangent in your own words (remember to include the name of the resource you used). Predict what you think it means for a circle to be tangent to the x-axis? Predict what you think it means for a circle to be tangent to the y-axis? You may change your predicitions after tomorrows class discussion.// //Tan= opposite/ adjacent, the opposite side side length of an angle divided by the adjacent side length of the angle.// http://mathworld.wolfram.com/Tangent.htm Tangent to the x-axis: To be (-y,0) Tangent to the y-axis: To be (0,-x)

Unit 2 Lesson 2 9/22/11


 * Unit 2 Lesson 1**

F(x) = x2 - 5
 * x || f(x) ||
 * -5 || 20 ||
 * -4 || 11 ||
 * -3 || 4 ||
 * -2 || -1 ||
 * -1 || -4 ||
 * 0 || -5 ||
 * 1 || -4 ||
 * 2 || -1 ||
 * 3 || 4 ||
 * 4 || 11 ||
 * 5 || 20 ||

g(x) = (x + 1)1/2 h(x) = 1/(x – 3)
 * x || g(x) ||
 * -5 || Error ||
 * -4 || Error ||
 * -3 || Error ||
 * -2 || Error ||
 * -1 || 0 ||
 * 0 || 1 ||
 * 1 || 1.4142 ||
 * 2 || 1.7321 ||
 * 3 || 2 ||
 * 4 || 2.2361 ||
 * 5 || 2.4495 ||


 * x || h(x) ||
 * -5 || -.125 ||
 * -4 || -.1429 ||
 * -3 || -.1667 ||
 * -2 || -2 ||
 * -1 || -.25 ||
 * 0 || -.333 ||
 * 1 || -.5 ||
 * 2 || -1 ||
 * 3 || Error ||
 * 4 || 1 ||
 * 5 || .5 ||


 * What type of function is f(x)? __//quadratic function because its highest power is in the form ax squared + bx - c.//__ g(x)? and h(x)? Explain.
 * What observations did you make about the table of values and graph of f(x)? Explain how this relates to the function and why you think this happened.__//It is a parabola because the value a is squared which causes the negative inputs to have a positive output except for some values because of b which is -5.//__
 * What observations did you make about the table of values and graph of g(x)? Explain how this relates to the function and why you think this happened. __//No values were below -1 because that would make the value of (x+ 1) negative and negative numbers don't have rational square roots.//__
 * What observations did you make about the table of values and graph of h(x)? Explain how this relates to the function and why you think this happened. //__The graph has two distinct components because x cannot be 3 since 3-3 is 0 and a number divided by 0 is undefined. Also as the graph has the asymptotes y= 0 and x = 3 because because as i explained x can never be 3 and also y can never be zero because the only way to get a quotient of 0 is if you are dividing 0.__//
 * Look up the mathematical definition for domain and write what domain means in your own words. How do your observations made about each function and table of values relate to this definition? Explain. //__A domain is the set of x values in a function.__//
 * What do you think would be an appropriate domain for a function representing the population of deer from the years 1975-2005? //__Explain. All real number greater than or equal to 1975 and less than or equal to 2005.__//


 * __ Unit 2 Lesson 10 __**
 * 1. In your own words, write the steps of performing a graphical transformation. Include any key reminders you think a students will forget in your description. **

If the new function has has + or - a number inside the brackets shift that number, n units left if it is positive and vice- versa, if the function has a = or - a number outside the brackets shift it n numbers up the graph if it is positive and vice versa. If the new function has a - attached to the x reflect it over the y- axis if it has a - on the outside reflect it over the x axis and stretch or shrink functions depending on the number if it now multiplied by a number.


 * 2. The graph of a function f(x) is illustrated. Use the graph of f(x) to perform the following graphical transformations. You do not need to show the shifted graph, you just need to list the 6 corresponding points. Answer each part seperately. **

(a) H(x) = f(x + 1) -2 (left 1 and down 2 units) (-1,2)(3,0)(-5,-2)(1,-1)(-3,0)(-7,-5)

(b) Q(x) = 2f(x) (stretched by 2)

(0,0)(2,12)(-6,6)(-2,4)(2,2)(-4,0)

(c) P(x) = -f(x) (reflection over x-axis)

(0,0)(-2,-2)(4,-2)(6,-2)(-4,0)(-6,3)

5 and -3 since the factored form of this quadratic would be y =(x - 5)(x + 3) and this equation would equal zero if 5 or -3 were placed in x.
 * 3. Suppose that the //x//-intercepts of the graph of f(x) are -5 and 3. Explain your thinking process or what helped you arrive at your answers. **

(a) What are //x//-intercepts if y = f(x+2)? (shifted to the left two units) -2 since this the equation y=x passes through the origin at 0 this equation would pass through at y=2 if x was zero so y= 0 if x was -2 (b) What are //x//-intercepts if y = f(x-2)? (shifted to the right two units) 2 2 since this the equation y=x passes through the origin at 0 this equation would pass through at y=-2 if x was zero so y= 0 if x was -2 (c) What are //x//-intercepts if y = 4f(x)? (stretched vertically by 4) 0 since the origin is at 0,0 and 0 times 4 is zero (d) What are //x//-intercepts if y = f(-x)? (reflected over the y-axis) 0 since y= x and x= 0 and -0 = 0
 * 4. Suppose that the function f(x) is increasing on the interval (-1, 5). Explain your thinking process or what helped you arrive at your answers. **

when the equation reaches the point -1,5 which is a minimum since it must decrease all the way down to this put before it increases making this point the turning point of the function to increasing.

(a) Over which interval is the graph of y = f(x+2) increasing? (left 2 units)

(b) Over which interval is the graph of y = f(x-5) increasing? (right 5 units)

(c) Over which interval is the graph of y = f(x)-1 increasing? (down 1 unit)

(d) Over which interval is the graph of y = -f(x) increasing? (reflected over x-axis)

(e) Over which interval is the graph of y = f(-x) increasing? (reflected over y-axis)

i am not understanding the question?

Unit 3 Lesson 5


 * __Summary Questions__**


 * 1. What do you notice about your solutions in part a and part b. In your explanation include what you got for solution to parts a and b to support your explanation.**


 * a) x squared - x - 1 with a remainder of -43**
 * b) -43**
 * The remainder of part a is identical to the answer of part b.**


 * 2. How can what you found be used as a short cut method to see if a number is a zero of a polynomial function or if a binomial is a factor before starting the synthetic division process? Explain.**


 * plug in k to the function you are dividing to find out if X- K has a zero since the opposite of k in a factored polynomial is a zero.**


 * 3. Look up the definition of the remainder theorem and factor theorem on page 215 and 216 of your text. Explain what these theorems mean in your own words using the examples above. Are there any restrictions to using the remainder theorem? Explain.**

Remainder Theorem states that when a polynomial f(x) is divided by the difference of x - k,the remainder is f(k).

Factor Theorem states that in a polynomial f(x) is a factor if f(k) = zero in terms of x-k.


 * 4. Explain when polynomial division is the appropriate method to use when dividing two polynomials. Explain when synthetic division is the most appropriate method to be used. Can you divide f(x) = 4x^3 - 8x^2 + 2x - 1 by g(x) = 2x + 1 using synthetic division? If you can explain what you would use as your k value.**

Polynomial long division can be used all the time while synthetic division can be used when the degree of x- k is 1. Synthetic division can be used on the equation above the k term would be -1/2.


 * __Unit 3 Lesson 9__**

1. The Fundamental Theorem of Algebra States: A polynomial function of a degree //n// has //n// zeros(real and non real). Some of these zeros may be repeated. Every polynomial of odd degree has at least one zero.


 * Explain what this statement means in your own words. In your description you should include an algebraic or graphical example to support your statement. You should also include the vocabulary of complex zeros, real zeros, and repeated zeros.**


 * The number of a power that a leading coefficient is raised to is the number of zeros it has whether those zeros are rational, or irrational repeated.**
 * eg in x squared the number of zeros is 2.**

2. Is it possible to find a polynomial with a degree of 3 with real number coefficients that has -2 as its only real zero? Explain.

yes if the value of q is positive and p is negative enough the equation may cross once.

3. The complex conjugate theorem states: Suppose that f(x) is a polynomial function with real coefficients. If a and b are real number with b not equal to zero and a + bi is a zero of f(x) then its complex conjugate a - bi is also a zero of f(x).


 * Explain what this statement means in your own words. You should include examples of complex conjugates when making your statement,.**


 * In a polynomial function when a and b are real numbers and b is not equal to zero and a + (b times the square root of -1) is a zero then**
 * a - (b times the square root of -1) is also a zero.**

4. Is it possible to find a polynomial function of a degree of 4 with real coefficients that has zeros 1+3i and 1-i. Explain.

Yes according to the complex conjugate theorem 1 - bi and 1 + i are also zeros giving a total of 4 zeros.

5. Is it possible to find a polynomial function of a degree of 4 with real coefficients that has zeros -3, 1 + 2i, and 1 - i. Explain.

yes the complex conjugate theorem holds that 1- 2i and 1 + i are also zeros.

__**Unit 4 Lesson 2**__

1. Create a rational function whose vertical asymptotes add to zero and whose zeros add to zero. Describe the asymptote behavior and end behavior of the function you created using limit notation. 1/x since x cannot equal zero because if x is zero 1/0 is irrational. asymptote- x = 0, y = 0 as x goes to 0- y= - and vice versa.

2. True or false: A rational function as a vertical asymptote at x = c every time c is a zero of the denominator. If the statement is false justify your answer using mathematical terminology learned in class and examples of at least 2 functions that make this statement false. True.

3. Describe how the graph of a nonzero rational function f(x) = (ax+b)/(cx+d) can be obtained from the graph y = 1/x.

Add a coefficient and variable x to it which will move the y coordinate by (x * coefficient) and up by b. cx stretches the graph while the d term moves the assymptote left or right.

4. Write a rational function with the following properties: (a) Vertical asymptotes: x = -5 and x = 2.

(b) Horizontal asymptote: y = -3.

1/(x+3)

(c) //y//-intercept 1.

1/(x squared + 1)

part 1 1) First switch x and y so you have x = y squared - 7. Then find the square root of both sides which is the square root of x = y minus the square root of 7.subtract the square root of x from both sides and do the same for y too. You get -y = the square root of x - the square root of 7. Then multiply it by -1 to get Y = X to the .5 power + the square root of 7.
 * __Unit 5 Lesson 4__**

2) Find a point and just switch the coordinates to find the corresponding inverse functions coordinates.

coordinate points: (5,-7) (4,-6) (3,-3) (1,9) inverses (-7,5) (-6,4) (-3,3) (9,1)

3) because it has a positive asymptote and cannot be negative. part 2 1) since the function log base 3 its inverse is 3 to the x power. 3 points of the log base 3 parent function are 1,0 3,1 and 9,2. To get the new function subtract 4 from the x values -3, -1 and 5. Raise Y to the -2 power: undefined, 1 and .25 then add 1 to them for the coordinates of -1,1 and 5,.25.

2) The vertical asymptote is the constant for the domain and the range is all real numbers.

Lessson 5

a)The square root of e two the power of 2 equals e. to get the reciprocal of e raise it the -1 power. the answer is -1

b) the answer is 3/4. It can be expanded into 1/4 log base 3 of 3 + 1/4 log base 3 of 9 which can be simplified into 1/4 of 1 + 1/4 of 2 which add up to .75

c)2 to the fourth power is 16. 4 times 3 is the answer 12.

d) natural logs of e cancel each other out. 2- 3 equals the answer -1.

e)to obtain 1/16 4 should be raised to the power of -.

f) expand the expression to get log base 5 of 2 minus log base 5 of 125 + log base 5 of 2. The log base 5 of 2s cancel out giving you log base 5 of 125 which is equals 3.

lesson 6

a) since division and subtraction are interchangeable expand it to log base 5 of 1then which equals zero. You are left with log -base 5 of 250. You can factor 250 out to 2 and 125. Since multiplication and addition are interchangeable you can simplify -log base 5 of 125 into -3 and -log base 5 of 2 is the remainder.

b)Expand it to natural log 8 + natural log 3 then expand 8 into 2 to the third power which can be written as 3ln 2.

lesson 7

a) 90 milligrams of the drug was administered to the patient after zero hours.

the function is set equal to 66. Subtract 90 from both sides you get -24 for Y so far. divide this by -52 to get .4615. enter the function and 24/52 into a calculator and find the intersection. After about .587 hours the amount of the drug in the persons system will be 66 milligrams.