Finding Slope Find the slopes of the lines passing through the following points. y y1 Formula for slope: m 1 m rise run Find the slopes of the lines passing through the following points. E #1: (7,0) and (0,4) E #: (, 5) and (1,9) E #3: (3, 5) and ( 1, 5) E #4: (7, ) and (7,5) Page 1
Equations of Lines Find the equation of a line given the slope and a point. Find the equation of the line with the given information. Write answers in slopeintercept form, if possible. You will need to know formulas: 1. Slope-intercept formula: y m b yy m( ). Point-Slope formula. 1 1 E. #1: m 5; through (,1) E. #: 3 m ; through ( 4, ) 5 Horizontal Equation: y number m 0 only has a y-intercept Vertical Equation: number m is undefined only has an -intercept E. #3: m 0; through ( 5,3) E. #4: m is undefined; through (, 7) Page
Find the equation of the line passing through the given points. y y1 1. Find the slope first. m 1. Pick one point and now use the Point-Slope formula. y y1 m( 1) 3. Write answers in slope-intercept form, if possible. E. #5: Passing through the points ( 1,3) and (4,7) E. #6: Passing through the points (3, 4) and ( 5, 1) Page 3
E. #7: intercept and y intercept 1 E. #8: Passing through the points (5, 6) and ( 3, 6) E. #9: Passing through the points ( 7, 4) and ( 7,8) Page 4
Graphing Lines Graphing lines using the slope and y-intercept. 1. Solve the equation for y.. Identify m and b. 3. Plot b on the y-ais. rise 4. From b, use the slope to get more points. run E. #1: 4 y 1 E. #: 3y 9 E. #3: 7 y E. #4: y E. #5: 3 1 0 Page 5
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Parallel Lines Parallel Lines have the same slope. Find the equations of the lines passing through the given points parallel to the given line. Write answers in slope-intercept form when possible. E #1: Through (,5); parallel to 3 7y 14 1. Find the slope of the given line by solving for y.. Use the slope and the given point to write equation of line. 3. Write answers in slopeintercept form when possible. E #: Through ( 4, 9) ; parallel to y E #3: Through (7, ) ; parallel to 8 Page 7
Perpendicular Lines Perpendicular Lines slopes are opposite reciprocals. (flip and change the sign) Find the equations of the lines passing through the given points perpendicular to the given line. Write answers in slope-intercept form when possible. E #1: Through (,5); perpendicular to y4 5 E #: Through ( 7,) ; perpendicular to 3 5y 15 1. Find the slope of the given line by solving for y.. Find the opposite reciprocal of the slope. We label this m 3. Use m and the given point to write equation of line. 4. Write answers in slope-intercept form when possible. E #3: Through ( 4, 9) ; perpendicular to y 8 E #4: Through (7, ) ; perpendicular to 3 Page 8
Midpoint and Distance Distance Formula: Use to find the distance between two points. distance ( ) ( y y ) 1 1 Eample 1: Find the distance between A(4,8) and B(1,1) Eample : Find the distance between A( 1,4) and B(3, ) Page 9
Midpoint Formula: Use to find the center of a line segment 1 y y1 midpoint, Eample 3: Find the midpoint of A(, 1) and B( 8,6) Page 10
Circles Circles Definition of Circle: A circle is a set of points in a plane that are located a fied distance, called the radius from a given point in the plane called the center. Standard Form of the Equation of a Circle: The standard form of the equation of a circle with the center ( hk, ) and the radius r is ( h) ( yk) r Eample 1: Write the equation of the circle in standard form given: Center (, 1) and r 4 Eample : Write the equation of the circle in standard form given: Center ( 5,3) and r 3 Page 11
Eample 3: Give the center and radius of the circle: ( 1) ( y4) 5 Eample 4: Give the center and radius of the circle: ( y3) 0 Eample 5: Give the center and radius of the circle: y y 8 6 0 Eample 6: Give the center and radius of the circle: y y 8 19 0 Page 1
Eample 7: Find an equation, in standard form, of the circle that satisfies the given conditions. a) Center (4, ) and tangent to the -ais. b) Center ( 3,5) and tangent to the y-ais. c) Center at the origin; passes through (4,1) d) Center is (6,7) ; passes through (1, 3) Page 13
e) Endpoints of the diameter are ( 8,1) and (,7) f) Endpoints of the diameter are ( 9, 8) and ( 3,0) Page 14
Review: y 9 Solvethesystembyeliminationmethod: 4 3y 14 WewillbeusingmatricestosolveSystemsofLinearEquations. Amatriisarectangulararrayofnumbersarrangedinrowsandcolumnsplacedinsidebrackets.The numbersinsidethebracketsofamatriarecalledelements. 13 15 1 Eample: 7 5 6 1 1 41 AnAugmentedMatriisamatrithatisusedtorepresentasystemofLinearEquations.Ithasa verticalbarseparatingthecolumnsofthematriintogroups(oneforthecoefficientsofthevariables inthelinearsystemandtheotherfortheanswers). onote:ifavariableismissing,weassignitthecoefficientofzero. RewritethefollowingsystemofequationsinanaugmentedmatriSystem SystemofEquations AugmentedMatri 3 y z 9 y 3z 3 y z 0 3 5z 1 4y z 5 3y 4 OnemethodtosolveasystemoflinearequationsusingamatriistogetthematriinEchelonForm, whichmeanstohaveonly1sinyourmaindiagonal(goingfromupperlefttolowerright)and0sbelow theones. OnceyouareinEchelonForm,youwillusebacksubstitutiontosolveyoursystem. 1 1 0 Eample: 0 1 3 0 0 1 Page 15
RowOperations Thereare3rowoperationsthatproducematricesthatrepresentsystemswiththesamesolutionset. RowOperation Notation Eample InterchangeRows R1 R3 1 1 3 3 3 18 1 1 1 1 0 MultiplyaRowbya R R 1 1 0 nonzeronumber 3 18 1 1 1 1 3 3 MultiplyaRowbya 1 R R R3 1 1 0 nonzeronumber 6 6 36 4 4 andthenaddthe producttoany 1 1 3 3 otherrow Tosolveasystemoflinearequationsusingmatriceswewould 1.Writesystemasanaugmentedmatri.UserowoperationstogetrowequivalentmatriinRowEchelonForm 3.UseSubstitutiontosolveforthevariables.Answersolutioninanorderedtriple. SystemofEquations 3 y z 9 Step1 y 3z 3 y z 0 Rewritesystemofequationsasaaugmentedmatri. 3 1 1 9 Step 1 1 3 3 1 1 0 Userowoperationstochangematritoarow 1 1 0 equivalentmatriinrowechelonform.... 0 1 3 Step3(thelongeststep) 0 0 1 Rewritematrifromrowechelonformtosystemof y z 0 equations. yz 3 Step4 z Solvethesystemusingbacksubstitution. (4, 1,) Step5 Page 16
SolvethefollowingSystemofEquations: 3 y z 9 1. y 3z 3 y z 0. y z 10 y z 6 3y z 15 Page 17
Review of Quadratic Formula The quadratic formula is derived from completing the square on the general equation: a b c 0 b b 4ac You MUST memorize the formula: a Process: 1. Write the equation in standard form: a b c 0. Identify ab,, and c. 3. Substitute numbers into formula. 4. Carefully do the arithmetic under the square root sign. 5. If possible, simplify the radical. 6. If possible, reduce the fraction. 1. 3 4 0. 8 5( 1) 3. 4 ( 1) 5 0 Page 18
Quadratic Types of Equations ReviewofFactoring: In previous math classes, you have learned to solve quadratic equations by the factoring method. 4 8 3 0 5 19 4 0 QuadraticTypesofEquations: We have equations that look like a quadratic, but have different eponents. Some eamples of these equations are: 4 4 8 3 0 1 3 3 5 19 4 0 1 6 7 3 0 Solve by factoring: 1. 4 4 8 3 0. 1 3 3 5 19 4 0 3. 1 6 7 3 0 4. 1 1 4 4 9 0 Page 19
Equations with Fractional Eponents ReviewofEponents: Remember that a fractional eponent can be written in radical form. 3 3 5 5 If you encounter an equation that has a variable raised to a fractional eponent, you solve it by raising both sides to the appropriate power. 3 3 5 5 Solve: 67 7 6. 3 9 5. 3 1 1 4 7. 1 7 1 18 Page 0
Functions A function is a set of ordered pairs where for every -value there is a unique y-value. The -values are called the domain (left to right). The y-values are called the range (bottom to top). The vertical line test can determine if a graph is a function or not. If a vertical line only crosses the graph once, then the graph is a function. Determine whether the following is a function. Give the domain and range of each relation. I {(10,8),(6, 4),(,0),(, 4)} K {(3,4),(3, ),(8,9),(1,0)} Properties of Graphs 1. Specific Values. Domain and Range 3. Intercepts-where the graph crosses the aes. Find the following: Use the graph to find: f ( ) f (3) For what value(s) of does f( ) 4? For what value(s) of does f( ) 0? Find the -intercept(s): Find the y-intercept: Page 1
Determine the domain, range, any intercepts and values. f ( ) =? f ( 1) =? f () =? Given f ( ) 5, evaluate each function at the given values and simplify answers. f ( 3) f () f( 1) Page
Given f ( ) 3 1, evaluate each function at the given values and simplify answers. f ( 3) f ( ) f( 5) Given h, evaluate each function at the given values and simplify answers. 1 4 h h 1 h Page 3
Properties of Functions Increasing/Decreasing Intervals: The part of the DOMAIN where y-values are increasing/ decreasing. Relative Maima: A point where a function changes from increasing to decreasing is called a relative maimum. Relative Minima: A point where a function changes from decreasing to increasing is called a relative minimum. With the given graph, 1. Determine the domain:. Determine the range: 3. Determine f ( 8) 4. Solve f ( ) 10 5. Find the intervals where the function is increasing: 6. Find the intervals where the function is decreasing: 7. Find the intervals where the function is constant: 8. Find the numbers at which f has a relative maimum: 9. Find the relative maima: 10. Find the numbers at which f has a relative minimum: 11. Find the relative minima: 1. Find all intercepts: 13. Find the values of for which f ( ) 0 14. Find the zeros of f Page 4
With the given graph, 1. Determine the domain:. Determine the range: 3. Find the intervals where the function is increasing: 4. Find the intervals where the function is decreasing: 5. Find the intervals where the function is constant: 6. Find the numbers at which f has a relative maimum: 7. Find the relative maima: 8. Find all intercepts: 9. Find the values of for which f ( ) 0 10. Find the zeros of f Page 5
Piecewise Functions A piecewise function is a function in which the formula used depends upon the domain the input lies in. We notate this idea like: formula 1 if domain to use formula 1 f( ) formula if domain to use formula formula 3 if domain to use formula 3 A cell phone company uses the function below to determine the cost, C, in dollars for g gigabytes of data transfer. 5 if 0 g Cg ( ) 5 10( g) if g Find the cost of using 1.5 gigabytes of data, and the cost of using 4 gigabytes of data. Evaluate each piecewise function at the given values. 1 if 0 f( ) if 0 a) f ( ) b) f () c) f (0) if 0 f( ) 1 if 0 a) f ( ) b) f () c) f (0) Page 6
The Difference Quotient The difference quotient is defined by: f ( h) f ( ), h 0 h Find the difference quotient of the following functions: 1. f ( ) 51. f( ) 6 3. f( ) 3 5 Page 7
Composite Functions f ( ) 3 1 f ( ) 5 6 g ( ) g( ) 5 6 1. f g (). g f () 1. ( f g)( 3). ( g f)( 5) 3. ( f g)(0) 4. ( g f )(3) 5. ( f g)(7) 6. ( g f )(9) Page 8
Find f g( ) and g f( ) 1. Given f ( ) 31and g ( ). Given f ( ) and g ( ) Find f g( ): 3. Given f( ) and 3 g ( ) Page 9
QuadraticFunctions Quadratic Function in General Form: Quadratic Function in Standard Form: Verte (or turning around point) = Ais of symmetry: Look at graph. these graphs where each verte is (0,0). Notice what the value of a does to the StepstoGraph: Find the verte. Determine the value of a. Plot the verte. From the verte, go over one unit (to the right and left) and then up or down a. Graph: f 1. Find the ais of symmetry, the domain, and the range. Page 30
Graph: f 1 3. Find the ais of symmetry, the domain, and the range. Graph: f 1 4. Find the ais of symmetry, the domain, and the range. Intercepts: How to find: Where the graph intersects each ais. Look at the graph does not always give eact answer Let 0 and solve for y Let y 0and solve for. Find the intercepts: f 1 4 Page 31
Find the intercepts: f 1 1 5 3 Find the intercepts: f 1 3 Page 3
QuadraticFunctionsinGeneralForm General Form: f ( ) a bc Standard Form: f( ) a( h) k Need to find the verte: ( hk, ) Use the formula: b h, k f( h) a f 6 7. Find the ais of symmetry, the domain, and the range. Graph: Find the intercepts. Page 33
f 4. Find the ais of symmetry, the domain, and the range. Graph: Find the intercepts. f 4 1. Find the ais of symmetry, the domain, and the range. Graph: Find the intercepts. Page 34
Quadratic Inequalities The standard form of a quadratic equation is: a b c 0 A quadratic inequality replaces the equal sign with inequality signs:,,, Process: 1. Write the given inequality in standard form.. Solve the corresponding equation by factoring or the quadratic formula. 3. Plot the answers on a number line. 4. Use test points or a graph to determine what interval solves the inequality. 5. Write answers in interval notation. 1. ( 1)( 5) 0. 35 0 3. 6 14 0 4. 35 0** 5. 1 0 6. 1 0 Page 35
Domain of a Function Finding the domain of a function: 1. The implied domain is the set of all real numbers for which the epression is defined. For all polynomial functions the domain is all real numbers or epressed in interval notation:, E. #1: f( ) 5 6 But what about rational functions? The question one must ask when finding the domain is Where is this function NOT defined?. Rational Functions: This is a function that is comprised of a ratio of polynomial functions. Thus, there is a denominator involved. We must always remember that DIVISION BY ZERO IS UNDEFINED!! To determine the domain of a rational function, set the denominator equal to zero and solve. These are the values that are NOT acceptable. E. #: f( ) 1 3 40 1 1 E. #3: f( ) 9 9 E. #4: f( ) 1 5 3 3. Radical Functions: This is a function that is underneath some type of radical. Remember when taking the EVEN root of a negative number, the answer is imaginary. Imaginary numbers are NOT acceptable for real valued functions. To determine the domain of a radical function, set the radicand greater than or equal to zero and solve. These are the values that ARE acceptable. E. #5: f( ) E. #6: f( ) 86 Page 36
E. #7: f( ) 3 5 E. #8: f( ) 3 6 E. #9: f( ) 5 4 Page 37
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Transformations: Graph Transformations g ( ) f( ) constant Moves the graph g ( ) f( ) constant Moves the graph g ( ) f( constant) Moves the graph g ( ) f( constant) Moves the graph g ( ) f( ) Multiplies all the y-values by g ( ) (Constant) f( ) Multiplies all the y-values by g ( ) a f( b) c a and c affects the y-values. b affects the -values Use the graph of y f( ) to obtain the following graphs: 1. g ( ) f( ). g ( ) f( ) Page 43
3. g ( ) f( ) 4. g ( ) f( ) 5. 1 g ( ) f( ) 6. g ( ) f( ) 7. 1 g ( ) f( 1) Page 44
8. f( ) 1 3 9. f 3 ( ) ( 1) 1 10. f 1 3 ( ) 3 Page 45
Piecewise Functions GraphingPiecewiseFunctions To graph a piecewise defined function, choose several values for each domain including the endpoints of each domain, whether or not that the endpoint is included in the domain. 1 if 0 1. f( ) 3 if 0. if 1 f( ) 1 if 1 3. 3 if f( ) 3 if Page 46
4. 3 if 4 0 f( ) if 0 if 0 5. 4 if 4 f ( ) if 4 1 3 if 1 Page 47
Graphs of Polynomial Functions In order to sketch a graph of a polynomial function, we need to look at the end behavior of the graph and the intercepts. The end behavior of the graph is determined by the leading term of the polynomial. 4 f ( ) 4 3 5 y f 3 ( ) 5 7 5 y f( ) 3(51) ( ) y Page 48
f ( ) ( 4) ( ) y Summary: If the leading coefficient has an even power, then the end behavior is the same: both up or both down If the leading coefficient has an odd power, then the end behavior will be opposites: one up and one down Intercepts: In order to find the y-intercept, set = 0 and solve for y. In order to find the -intercept, set y = 0 and solve for. f ( ) ( ) ( 1) f 3 ( ) 3 3 Page 49
Multiplicity of the -intercepts 1. Multiplicity of 1 or single: the graph passes through the -ais like a line.. Multiplicity of or even: the graph passes bounces off the -ais like a parabola. 3. Multiplicity of 3 or odd: the graph squiggles through the -ais like a cubic function. Graphthefollowing: End Behavior, y-intercept, the multiplicity of the -intercepts. 3 f( ) ( ) ( 1)( 3) y f ( ) 3 3 4 3 y Page 50
f 4 ( ) 4 3 1 y y f( ) (31) ( 1) 3 f ( ) 4 1 y Page 51
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Synthetic Division and Remainder Theorem Synthetic Division is a condensed method of long division. It is quick and easy. Unfortunately, it can only be used when the divisor is in the form of ( a) Synthetic division: 1. 9 5 1 1. 3 15 5 3. 3 5 4 1 Reminders: 1. Write both polynomials in standard form.. Fill in all missing terms with a place holder of zero. 3. Write your answer as a polynomial that is one degree less than the dividend (numerator). RemainderTheorem: When dividing a polynomial by ( c), then the remainder is f () c. 3 If f( ) 5 4, find f (1). (refer to #3) If 3 f( ) 11 7 19, find f ( 1). Page 53
If 4 3 f( ) 1 6 5, find f 3. FactorTheorem: Let f ( ) be a polynomial. Then if f () c 0, then ( c) is a factor of f ( ). And if ( c) is a factor of f ( ), then f () c 0. (refer to #) 3 Solve the equation 3 116 0, given that 3 is a zero (or factor) of the function 3 f( ) 3 11 6 4 3 Solve the equation 3 17 19 118 0, given that 3 is a zero of multiplicity of 4 3 two of the function f( ) 3 17 19 1 18 Page 54
Zeros of Polynomial Functions Some polynomials cannot be factored by traditional methods. The Rational Zero or Root Theorem gives us another method to find the -intercepts or zeros of a polynomial. The theorem states that a list of possible rational zeros of a polynomial can be found by taking the factors of the constant term (p) and dividing them by the factors of the leading coefficient (q). Make a list of possible rational zeros: 3 1. f( ) 4 15. f 5 ( ) 10 5 After making the list, we can use it along with synthetic division and the graph of the function to try and factor the polynomials and find the roots (zeros). f 3 ( ) 3 5 4 4 Page 55
f 4 3 ( ) 6 11 1 3 f 4 3 ( ) 7 5 13 3 Page 56
f 4 3 ( ) 3 3 16 1 f 5 3 ( ) 8 8 1 Page 57
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Graphing Rational Functions A rational function is the ratio of two polynomial functions. In order to sketch a graph, we must find all the intercepts and all the asymptotes. Process for sketching a graph of a rational function: 1. Find the y-intercept by setting = 0.. Find the -intercept(s) by setting y = 0. 3. Find the vertical asymptotes by setting the denominator = 0. (The denominator of the reduced function.) 4. Find the horizontal asymptote (if one eists) by comparing degree of the numerator to the degree of the denominator. 5. Plot the intercepts and graph the asymptotes. Plot a few additional points to complete the graph. 3 1. f( ) y-intercept: -intercept(s): Vertical Asymptote(s): Horizontal Asymptote: If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote will be: coefficient of the leading term of the numerator y coefficient of the leading term of the denominator Page 59
f( ). y-intercept: 3 6 -intercept(s): Vertical Asymptote(s): Horizontal Asymptote: If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote will ALWAYS be 3. f( ) y-intercept: 4 1 -intercept(s): Vertical Asymptote(s): Horizontal Asymptote: Page 60
Polynomial Inequalities Process: 1. Write the given inequality in standard form: eponents in descending order and zero on the right hand side.. Solve the corresponding equation by factoring. 3. Plot the answers on a number line. 4. Use test points OR a graph to determine what interval solves the inequality. 5. Write answers in interval notation. 1. ( 4)( 1)( 7) 0. ( 3)( 4)( 1) 0 3. ( ) ( 4)( 6) 0 3 4. 18 30 8 0 5. 3 4 4 0 6. 3 5 9 45 0 Page 61
Rational Inequalities Process: 1. Make one side of the inequality zero.. Combine all of the terms on the non-zero side into a single fraction. 3. Set both the numerator and denominator EQUAL to zero and solve these equations. These are the boundary points. 4. Plot these points on a number line. 5. Look at corresponding graph and shade either above or below -ais. 6. Write answers in interval notation. 3 1. 0. 3 1 4 Page 6
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Inverse Functions Review:DefinitionofaFunction A function is a set of ordered pairs where for every -value there is a unique y-value. Graphically: Use the vertical line test to determine if the graph is a function. Determine if the following graphs are functions? y y y OnetoOneFunctions(1 1functions) A function is said to be one to one if each y-value corresponds to only one -value. Graphically: Use the horizontal line test to determine if the following functions are One to One Functions. Determine if the following funcitons are One to One Functions y y y WhatisanInverseFunction? Only one-to-one functions have inverse functions. A function and its inverse can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs some operation on this value, and creates an output answer. The inverse of this function takes the output answer, performs some operation on it, and arrives back at the original function's starting value. Page 64
f ( ) 3 The inverse of f ( ) 3? Domain of f Range of f DefinitionofInverseFunctions: If f is a one to one function, then f 1 ( ) is the inverse function of f if: 1 1 f ( f Note: f ( )), for every in the domain Pronounced: f inverse of. And f 1 ( f( )), for every in the domain 1 f VerifyingInverses: Find f ( g ( )) and g( f( )) to determine whether each pair of functions f and g are inverses of each other. 9 1. f( ) and g ( ) 4 9 4 is notation for the inverse of f. is NOT f to the negative 1 eponent. 3 f( ) and g ( ) 3 Page 65
3. f( ) 5 and g ( ) 5 FindingtheInverseFunction Process: 1. Replace f ( ) with y.. Switch and y. 3. Solve for y. 4. Re-write y as f 1 ( ). For the following problems: a) Find the inverse of the given function. b) VERIFY your equation is the inverse by showing 1 f ( f ( )) and f 1 ( f( )). 1. f ( ) 3 1. f 3 ( ) 4 Page 66
3. f( ) 5 4. 4 f( ) 9 5. f( ) 3 5 Page 67
GraphingInverseFunctions In order to graph the inverse of a function, you need to switch the domain and the range. In other words, reverse the order of the ordered pairs. Page 68
An eponential function is a function where a positive number is raised to a power. f ( ) b where b > 0 and b 1 Eponential Functions NOT Eponential Functions Reviewofeponents: 0 1 1 a b 1 GraphthefollowingEponentialFunctions: f( ) 1 f( ) Domain: Range: X-int: Y-Int: Asymptote: Domain: Range: X-int: Y-Int: Asymptote: Page 69
Transformations f ( ) b All eponential functions (in basic form) have point on their graphs: (1, ) (0, ) Vertical Shift f ( ) b c f ( ) b c f ( ) b ( c) Horizontal Shift f ( ) ( c) b Reflections f ( ) b f ( ) b Vertical Stretch and Compress f ( ) c b Note that cb ( cb) TransformationsEamples: Sketch f( ) 3 Then, sketch the following 4 f( ) 3 f( ) 3 4 f( ) 3 f( ) 3 Page 70
NaturalNumbere: Of all possible choices of bases, the most preferred or most natural base it the number e. The number e has important significance in science and mathematics. It is often called Euler s number named after Leonhard Euler. e.71881884590453536087471357... The number e is defined by: If n is a positive integer, then 1 1 n n e as n (discussed in Calculus) NOTE: The number e is a number, not a variable. Sketch f ( ) e Then, sketch the following f ( ) 4 e f( ) e 3 f ( ) 3e f 1 ( ) e Page 71
Review: Find the inverse of f( ) 3 Find the inverse of f( ) The eponential function has an inverse called DefinitionofLogarithmicFunction: y y For 0 andb0, b 1, b is equivalent to log b f is the logarithmic function with base b. The function ( ) log b GraphthefollowingInverseFunctions: f( ) f ( ) log Domain: Range: X-int: Y-Int: Asymptote: Domain: Range: X-int: Y-Int: Asymptote: Page 7
GraphthefollowingInverseFunctions: f( ) 4 f ( ) log4 Domain: Range: -int: y-int: Asymptote: Domain: Range: -int: y-int: Asymptote: Note: Some bases are used frequently, and have simplified notation. Common Log (Base 10): log 10 = Natural Log (Base e): log e = Inverse Functions Logarithm Form Eponential Form f ( ) b f( ) 10 f ( ) e Transformations: Parent Function: f log b All logarithmic functions (in basic form) have points on their graphs: (1, 0) and ( b,1) Vertical Shift f log b c f log b c Horizontal Shift f log b c f log b c Reflections f log b Vertical Stretch and Compress f clog b Page 73
Sketch f ( ) log Then, sketch the following f ( ) log f ( ) log( ) f ( ) log( 1) 4 f ( ) log f ( ) log( ) 3 Page 74
Sketch f ( ) ln Then, sketch the following f( ) ln 3 f ( ) ln 3 f ( ) ln( ) 4 f ( ) 1 ln f ( ) ln( 1) Page 75
Properties of Logarithms Review of eponent properties: 3 3 11 0 5 7 1 6 Compare eponents to logarithms: Eponential Form Log Form 3 8 3 1 9 81 9 3 7 3 log16 4 log 3 1 1 3 Evaluate the following epressions: log7 49 log3 7 1 1 log 9 log6 3 6 log6 6 log 3 1 3 log819 log11 Common logarithms: Base Ten log10 5 log 10 Page 76
Natural Logarithms: Base e ln e 1 ln e ln e Since logs and eponents are inverse functions, they undo one another. The following properties show this: log a a and log a a log5 5 7 3 log ln e 7 5 ln 8 log e 411 4 log1000 PropertiesofLogarithms For 0 M and N 0: 1. Product Rule: log M log N log MN b b b. Quotient Rule: log M log N log b b b 3. Power Rule: plog M log M b b p Use the properties of logs to epand each epression as much as possible. M N log a p 4 qr 3 log 5 a u3 v Epress as a single logarithm: 3 3loga loga y loga z log a log a( 3) log a( 3) Page 77
Eponential Equations An eponential equation has the variable in the eponent. The easiest way to solve y is to use the same base. ( a a ) Process: 1. Isolate the base.. Re-write each side of the equation with the same base. 3. Equate the eponents. 4. Solve. Solve the following: 16 1 4 64 16 8 1 3 1 49 7 e 3 1 3 e 1 9 7 Page 78
An eponential equation has the variable in the eponent. When you cannot get the bases to be the same, you have to use logarithms to solve. We make use of the following logarithmic property: p plog M log M or pln M ln M a a Process: 1. Isolate the base.. Take the log (ln) of both sides of the equation. 3. Use the log property (above) to re-write the eponents as coefficients. 4. Solve. 5. Use a calculator to approimate the solution. p Solve the following: 7 60 1 40 8 1 e 7 13 3 5 11 35 4 3 1 5 3 7 1 5 Page 79
Logarithmic Equations Form 1 logbm logb N constant 1. Combine logs using log properties.. If the coefficients do not equal to one, then use the log property: p plogbm logbm. Now your equation should have the form of: log D b constant constant 3. Re-write the equation in eponential form to get rid of the log: b 4. Solve for variable. 5. Check your answer back into the original equation. (You can only take the log of numbers that are greater than zero.) D Solve the following: log (3 ) ln( 4 ) 3 5 log( ) log( 1) 1 log ( 3 ) log ( ) Page 80
Logarithmic Equations Form logbm logb N logbc 1. Combine logs using log properties.. If the coefficients do not equal to one, then use the log property: p plogbm logbm. Now your equation should have the form of: logbd logbc 3. Using property of equality, you can now say that D C 4. Solve for variable. 5. Check your answer back into the original equation. (You can only take the log of numbers that are greater than zero.) log log ( 5) log 6 ln( 4) ln( 1) ln log log7 log100 log 3(1) log 3( 3) log 3( 5) ln( 4) ln( 1) ln(3 1) Page 81