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E - P o r t f o l i o

E - P o r t f o l i o. By: Student 3 Pre-AP Algebra II 7 th period. END. Parent Function 1. Parent Function 2. Calculator Tips. bibliography. Parent Function 3. Parent Function 4. Home Page. Word Problem. Home Page: Parabolic Functions. Parent Function. Translation: Horizontal.

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E - P o r t f o l i o

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  1. E-Portfolio By: Student 3 Pre-AP Algebra II 7th period

  2. END Parent Function 1 Parent Function 2 Calculator Tips bibliography Parent Function 3 Parent Function 4 Home Page Word Problem

  3. Home Page: Parabolic Functions Parent Function Translation: Horizontal Translation: Vertical Reflection How to Graph

  4. Home Page: Logarithmic Functions Parent Function Translation: Horizontal Translation: Vertical Reflection How to Graph

  5. Home Page: Square Root Functions Parent Function Translation: Horizontal Translation: Vertical Reflection How to Graph

  6. Home Page: Rational Functions Parent Function Translation: Horizontal Translation: Vertical Reflection How to Graph

  7. Home Page: Word Problem Word Problem Step by Step Step By Step: Predictions Graph

  8. y x Parent Function 1: Parabola Formal Definition of a Parabola: A parabola is the set of all points in the plane equidistant from a given line L (the conic section directrix) and a given point F not on the line (the focus). Basically, this means that a parabola is the graph of a quadratic function, and it is U-shaped (meaning it will either open up, open to the right, open to the left, or simply flip upside down) over the y-axis.   DOMAIN: (-∞,∞) RANGE: (0,∞)

  9. y y x x Translation: Horizontal Translations: To The Right To The Left DOMAIN: (-∞,∞) RANGE: [0,∞) DOMAIN: (-∞,∞) RANGE: [0,∞)

  10. y y x x Translation: Vertical Translations: Down Up DOMAIN: (-∞,∞) RANGE: [2,∞) DOMAIN: (-∞,∞) RANGE: (-2,∞)

  11. Which is a reflection of: Over the x-axis Reflection DOMAIN: (-∞,∞) RANGE: (0,∞)

  12. How to Graph • Find the focus and directrix by: • Putting the equation into standard form x² = 4cy or y² = 4cx 2. Identifying c (the distance from the vertex to the focus and directrix) • MAKE A TABLE OF VALUES WITH A MINIMUM OF THREE POINTS, WITH THE VERTEX AS ONE OF THE POINTS, AND TWO ON EITHER SIDE. • GRAPH THE DIRECTRIX (A DOTTED LINE), THE FOCUS (a single point), AND THE points on the TABLE OF VALUES. • CONNECT THE DOTS FROM THE TABLE OF VALUES. • Your graph is done!!

  13. y x Parent Function 2: Logarithm Formal Definition of a Logarithmic Function: The power to which a base, such as 10, must be raised to produce a given number. If n^x= a, the logarithm of a, with n as the base, is x; symbolically, log(base)n (a) = x. For example, 10^3 = 1,000; therefore, log10 1,000 = 3. The kinds most often used are the common logarithm (base 10), the natural logarithm (base e), and the binary logarithm (base 2). DOMAIN: (0,∞) RANGE: (-1.5,∞)

  14. Translations: To The Left To The Right y y x x Translation: Horizontal DOMAIN: [-3,∞) RANGE: [-1.5,∞) DOMAIN: [3,∞) RANGE: [-1.5,∞)

  15. y Translations: Down Up y x x Translation: Vertical DOMAIN: [0,∞) RANGE: [.5,∞) DOMAIN: [0,∞) RANGE: [-3.5,∞)

  16. Which is a reflection of: Over the x-axis Reflection DOMAIN: (0,∞) RANGE: (-1.5,∞)

  17. How to Graph There are two ways to graph a logarithmic function • Option 1 • Find the inverse of the equation: to • Graph the inverse (table of values, 3 points, etc.) • Reflect the inverse over the line (into the first and fourth quadrant only) • Option 2 • Make a table of values with a minimum of three points, staying within the first and fourth quadrants) • Graph the points • Your graph is done!!

  18. Parent Function 3: Square Root Formal Definition of a Square Root Function: The graph of a square root. Formal Definition of a Square Root: If b^2 = a, then b is a square root of a y x DOMAIN: (0,∞) RANGE: [0,∞)

  19. Translations: To The Right To The Left y y x x Translation: Horizontal DOMAIN: [3,∞) RANGE: [0,∞) DOMAIN: [-3,∞) RANGE: [0,∞)

  20. y Translations: Up Down y x x Translation: Vertical DOMAIN: [0,∞) RANGE: [3,∞) DOMAIN: [0,∞) RANGE: [-3,∞)

  21. Which is a reflection of: Over the x-axis Reflection DOMAIN: (0,∞) RANGE: (0,∞)

  22. How to Graph There are two ways to graph a square root function • Option 1 • Find the inverse of the equation: to • Graph the inverse (table of values, 3 points, etc.) • Reflect the inverse over the line (into the first quadrant only) • Option 2 • Make a table of values with a minimum of three points, staying within the first quadrant) • Graph the points • Your graph is done!!

  23. Parent Function 4: Rational Formal Definition of a Rational Function: A function in the form of f(x) = p(x) , q(x)≠0 q(x) x y DOMAIN: (-∞,0) (0,∞) RANGE: (-∞,0) (0,∞)

  24. y y Translations: To The Left To The Right x x Translation: Horizontal DOMAIN: (- ∞,-3) (-3,∞) RANGE: (-∞,0) (0, ∞) DOMAIN: (- ∞,3) (3,∞) RANGE: (-∞,0) (0, ∞)

  25. y y Translations: Down Up x x Translation: Vertical DOMAIN: (- ∞,0) (0,∞) RANGE: (-∞,-3) (-3, ∞) DOMAIN: (- ∞,0) (0,∞) RANGE: (-∞,3) (3, ∞)

  26. Which is a reflection of: Over the x-axis Reflection DOMAIN: (-∞,0) (0,∞) RANGE: (-∞,0) (0,∞)

  27. How to Graph 1. Find the asymptotes and the holes of the graphs. REMEMBER: BOBOBOTNEATSDC for the horizontal asymptotes BOBO: Bigger on bottom, its is zero BOTN: bigger on top, there isn’t any EATSDC: if the exponents are the same, divide the coefficients For the vertical asymptotes, set the expression in the denominator equal to zero. 2. Graph the asymptotes 3. Make a table of values: at least three points on either side of the asymptote. (If there are more than one, graph three points on either side, and three points in between each two asymptotes) 4. Graph the table of values, remembering to not cross the asymptotes. 5. Your graph is done!!

  28. Word Problem An RC circuit with resistance (R) of 10,000Ωand capacitance (C) of 10 µF is fully charged at 10 volts. The equation for voltage discharge is: Where R is in ohms (Ω), C is in farads (F) and V is the initial “impressed voltage” in volts. • What is the circuit voltage after .1 seconds? PREDICTION: • If, for this circuit, .0005 is considered zero, how long will it take for the circuit to discharge completely?

  29. Step By Step To solve the first question: I. Write down the equation, and then plug in the numbers Where t is in seconds II. Solve for V(t)

  30. Step By Step: Predictions To Predict the second question I. Again, Write down the equation and plug in the numbers Where t is in seconds II. Solve for t

  31. 7 6 5 4 3 2 1 0 .4 .1 .2 .3 .5 .6 .7 .8 .9 Graph Graph intersects the y-axis at 10 volts. Graph never intersects the x-axis. VOLTS TIME

  32. CalculatorTips To graph a function on a TI83 calculator, follow these steps: 1. Press the ON button to turn the calculator on. 2. Press the y= button. The screen should look like this: 3. Type the equation into y sub 1 4. Hit WINDOW and set the dimensions that would best fit the equation. 5. Press GRAPH. The graph of the function should show up onto the screen. 6. To see a table of values for the function, hit the 2ND button, and then hit GRAPH again. The table should show up on the screen. #2 #3 #4 #5 #6

  33. Bibliography • No Author. "3 Entries Found for Logarithm." Dictionary.Com. 2006. The American Heritage® Dictionary of the English Language, Fourth Edition. 7 May 2006 <http://dictionary.reference.com/search?q=logarithm>. • Ward, Leslie. "Parabolas." Mrs. Albrecht's RHHS Classroom. 16 Nov. 2005. Mrs. Albrecht's class projects. 7 May 2006 <http://rhhs.rockwallisd.com/calbrecht/parabolas.pub>.

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