Nonlinear Functions. = if x<0. x x
|
|
- Janice Walton
- 7 years ago
- Views:
Transcription
1 Nonlinear Functions By definition, nonlinear functions are functions which are not linear. Quadratic functions are one type of nonlinear function. We discuss several other nonlinear functions in this section. A. Absolute Value Recall that the absolute value of a real number is defined as = if 0 if <0 Consequently, the graph of the function f ( ) = is made up of two different pieces. For > 0, the graph is the graph of the linear function, and for < 0, the graph is the graph of the linear function. We show the three different graphs below. f ( ) The graph of g( ) = for > 0 f ( ) graph across the ais. = for < 0 f ( ) = = can be obtained from the graph of f ( ) = by reflecting the Reflecting f ( ) = across the ais to obtain g = the graph of ( )
2 We give the graphs of f ( ) = and g( ) = together below for easy reference. f ( ) = g( ) = The graphs of f ( ) of f ( ) =. = and g( ) = above can be used to graph shifts and scalings Eample: The graph of the function g ( ) = 3 can be given in 3 steps. We start by graphing the function f ( ) = by scaling the graph of by a factor of. Then we shift this graph unit in the direction. Finally, we shift -3 units in the y direction. The process is demonstrated below. Scaling by. Shifting unit in the direction. Shifting 3 units in the y direction. Eample: The graph of the function g ( ) = + can be given in steps. We start by graphing the function f ( ) = by scaling the graph of by a factor of /. Then we shift this graph - unit in the direction.
3 Scaling by -/. Shifting - unit in the direction. Eample: We can graph the function g ( ) = as a shift and scaling of the function f ( ) =. First, note that = =. So, we can rewrite g in the form g ( ) =. Now we can recognize that g is the result of scaling the function f ( ) = by a factor of, and then shifting the result ½ units in the direction. The result is shown below. Scaling f ( ) = by a factor of. Shifting the result ½ units in the direction. Eample: We can see from the eample above that the graph of the function g ( ) = is made up of two lines. The equations for these line, along with the restrictions on can be found by using the definition of absolute value. First, recall that a a if a 0 = a if a < 0 Consequently,
4 if 0 if 0 = = ( ) if < 0 + if < 0 We can solve the inequality 0 as follows. 0 If we combine this information with the definition of g, we have if / g ( ) = = + if < / So, g is a piecewise function, made up of two linear functions. More precisely, and Remark: A function written in the form g ( ) = if / g ( ) = + if < / a + b if c G ( ) = d + e if < c is said to be a piecewise linear function. Notice that the function in the eample above is an eample of a piecewise linear function. In that eample, the values a, b, c, d and e are given by, -, ½, - and respectively. Eercises:. Describe how to obtain the graph of the function f ( ) = 3 from the graph of Q ( ) =. Then graph the function.. Describe how to obtain the graph of the function F( ) = from the graph of Q ( ) =. Then graph the function. 3. Describe how to obtain the graph of the function h ( ) = from the graph of Q ( ) =. Then graph the function. 4. Describe how to obtain the graph of the function G ( ) = + from the graph of Q ( ) =. Then graph the function.
5 5. Describe how to obtain the graph of the function R ( ) = + from the graph of Q ( ) =. Then graph the function. 6. Describe how to obtain the graph of the function T( ) = + 3 from the graph of Q ( ) =. Then graph the function. 7. Describe how to obtain the graph of the function H( ) = 3 from the graph of Q ( ) =. Then graph the function. 8. Describe how to obtain the graph of the function H( ) = + 4 from the graph of Q ( ) =. Then graph the function. 9. Use the definition of absolute value to write the function G ( ) = + as a piecewise linear function. 0. Use the definition of absolute value to write the function T( ) = + 3 as a piecewise linear function.. Use the definition of absolute value to write the function R ( ) = + as a piecewise linear function.. Use the definition of absolute value to write the function H( ) = 3 as a piecewise linear function. 3. Use the definition of absolute value to write the function M( ) = + 4 as a piecewise linear function. B. Polynomial Functions We have already seen some special types of polynomial functions. A linear function f ( ) = m+ b is a first degree polynomial function. A quadratic function g( ) = a + b + c is a second degree polynomial function. In general, an is a function of the form where { 0,,,... } Eamples: n F( ) = a + + a + a n 0 n and a,..., n a, a0 with an 0. th n degree polynomial function
6 . f( ) = is a polynomial of degree H( ) = is a polynomial of degree R( ) = 3 + is a polynomial of degree. 7 4 F( ) = is a polynomial of degree The graphs of polynomial functions can sometimes be very complicated. For eample the 7 4 graph of F( ) = 4+ is shown below. 00 F = ( ) 4 One class of polynomial functions which have predictable graphs is given by where { 0,,,... } F( ) = n n. If n is an even natural number, then the graph of F( ) = has a graph that is similar to the graph of f ( ) =. n
7 n F( ) = when n is even. Although these graphs are similar, they are not called parabolas. The graph of f ( ) 3 = is shown below along with a table of values for the function n If n 3 is an odd natural number, then the graph of F( ) = has a graph that is similar 3 to the graph of f ( ) =.
8 Eample: The graph of P ( ) ( ) 3 by - unit in the direction. = + can be obtained by shifting the graph of f ( ) = 3 4 Eample: The graph of P ( ) = + can be obtained by shifting the graph of by unit in the y direction. f ( ) = 4 Eample: The graph of g ( ) process is shown below. P 4 ( ) 8 = + can be obtained by reflecting the graph of 4 = across the ais, and then shifting the result 8 units in the y direction. The
9 g ( ) 4 = Reflection of g ( ) 4 = across the ais. Shifting the result 8 units in the y direction. 4 P ( ) = + 8 Eercises: 4. Use the definition of polynomial function to identify that P ( ) = + 8 is a polynomial function. Be sure to give the degree of this polynomial function. 3. Use the definition of polynomial function to identify that F( ) = is a polynomial function. Be sure to give the degree of this polynomial function. 3. Use the definition of polynomial function to identify that f ( ) = 3 + is a polynomial function. Be sure to give the degree of this polynomial function. 4. Use the definition of polynomial function to identify that 5 3 h ( ) = is a polynomial function. Be sure to give the degree of this polynomial function Graph the polynomial function f( ) = Graph the polynomial function g ( ) = Graph the polynomial function h ( ) = ( ) Graph the polynomial function F ( ) 3 ( ) = Describe how to obtain the graph of the function f ( ) 7 graph of g ( ) 7 =. ( ) = + 3 from the
10 0. Describe how to obtain the graph of the function f ( ) graph of g ( ) =.. Describe how to obtain the graph of the function ( ) 7 graph of g ( ) 7 =. Hint: 4+ = ( ) = from the f( ) = 4+ + from the C. Rational Functions Rational functions are functions which can be written in the form P ( ) f( ) = Q ( ) where P ( ) and Q ( ) are polynomial functions. Eamples:. f ( ) = 3 + is a rational function since it can be written as a quotient of the nd degree polynomial function P ( ) = 3 + and the 0 th degree polynomial function Q ( ) =. In general, all polynomial functions are rational functions.. f( ) = is a rational function since it is the quotient of polynomials with P ( ) = and Q ( ) = Remark: The domain of a rational function is the set of all values of where the denominator Q() is nonzero. The graphs of rational functions can be very complicated. For eample, the graph of 3 4 f( ) = is given below. 3 4
11 We can learn a lot about rational functions by studying the simple rational function g ( ) =. Notice that we can not evaluate g at = 0, but we can evaluate g at every other value of. Consequently, the domain of g is { 0}. We can also say something about the behavior of g. First, notice that when is positive and very close to zero, the value of / will be very large. We can see this in the following table of values / Notice that the closer gets to 0, the larger the values of g() become. In fact, the values of g() can be made larger than any positive value by choosing appropriate positive values of sufficiently close to 0. In mathematical terms, we say that We write this symbolically as g ( ) goes to as approaches 0 through positive values g ( ) as 0 Similarly, notice that when is negative and very close to zero, the value of / will be very large in the negative sense; i.e. / will be negative and / will be very large. We can see this in the following table of values / Notice that the closer that negative values of get to 0, the larger the values of g() become in the negative sense. In fact, the values of g() can be made larger in the negative sense than any negative value by choosing appropriate negative values of sufficiently close to 0. In mathematical terms, we say that +
12 g ( ) goes to as approaches 0 through negative values We write this symbolically as g ( ) as 0 We can show this information graphically as shown below. This type of behavior is typical of rational functions, and other more eotic functions. In general, if f is a function and a is a real number, we say f has a vertical asymptote at = a if and only if at least one of the following occur: f ( ) as a, f ( ) as a, f ( ) as a + or f ( ) as a + P ( ) A general rational function f( ) = will have a vertical asymptote at a value = a Q ( ) where Qa ( ) = 0 and Pa ( ) 0. The specific rational function g ( ) = has a vertical asymptote at = 0. Notice that we can also say something about the behavior of g ( ) = when is a large positive number. In this case, / will be a very small positive number. X / In general, we can make g() smaller than any positive number by choosing to be a sufficiently large positive number. Notice that g() is positive whenever is positive. Combining this information, we say
13 We write this symbolically as g ( ) goes to 0 through positive values as approaches + g ( ) 0 as Similarly, we can say something about the behavior of g() when is a large negative number (i.e. large in the negative sense). In this case, / will be a negative number which is very close to / In general, we can make g() closer to 0 than any negative number by choosing sufficiently large in the negative sense. Notice that g() is negative whenever is negative. Combining this information, we say g ( ) goes to 0 through negative values as approaches We write this symbolically as g ( ) 0 as We incorporate this information into our earlier picture below. P ( ) This type of behavior is typical of more general rational functions f( ) = Q ( ) whenever the degree of P ( ) is less than or equal to the degree of Q. ( ) In general, if c is a real number, we say f has a horizontal asymptote at y = c if and only if at least one of the following occur:
14 The function g ( ) f( ) c as or f( ) c as = has a horizontal asymptote at y = 0. P ( ) A general rational function f( ) = will have the horizontal asymptote y = 0 if Q ( ) and only if the degree of P ( ) is less than the degree of Q. ( ) In the case when the degree of P ( ) is equal to the degree of Q, ( ) if n P ( ) = a + + a+ a n 0 and n Q ( ) = b + + b+ b n 0 are polynomial functions of degree n, then an asymptote y =. b n P ( ) f( ) = will have the horizontal Q ( ) We plot a set of data points below to obtain a graph of g ( ) =. Notice how the behavior of the graph incorporates our comments from above. /
15 We can see this more clearly if we plot a more complete set of values. Eample: The graph of ais. f( ) = is the reflection of the graph of g ( ) = across the -/ is the reflection of / across the ais Every rational function of the form a + b f( ) = c + d
16 (with c 0 ) is a scaling, reflection, and/or shift of the following series of algebraic manipulations: g ( ) a + b f( ) = c + d a c b = + c c+ d c+ d a c+ d a d b = + c c+ d c c+ d c+ d a bc ad = + c c c+ d a bc ad = + c c + d/ c =. We can see this by making The calculations above show that the graph of f can be obtained by shifting the graph of bc ad g ( ) = by d/c units in the direction, then scaling the result by, and finally c bc ad shifting by a/c units in the y direction. Notice that if < 0 then the scaling also c incorporates a reflection across the ais. This function will have a vertical asymptote at = d/c and a horizontal asymptote at y = a/c. + 3 Eample: Consider the rational function f( ) =. Notice that the denominator is 4 zero when 4= 0, and this occurs when =. Also, notice that the numerator is not zero at this value of. As a result, f has a vertical asymptote at =. Also, in this case, the polynomials in the numerator and denominator have the same degree (namely ). So, from the leading coefficients, we know that f has a horizontal asymptote at y = ½. Finally, we can see that
17 + 3 f( ) = 4 3 = = ( )( 3) = = + + Therefore, the graph of f can be obtained from the graph of g ( ) = by scaling by a factor of 5/, shifting the result by - units in the direction, and then shifting ½ units in the y direction. We show the graph below along with dotted lines for the vertical and horizontal asymptotes. Notice that in this case, the presence of the vertical and horizontal asymptotes are due to the behavior given by + f( ) as, f( ) as, f( ) as and + f( ) as. Remark: We can see the occurrence of the horizontal asymptote in the eample above by making the following algebraic simplification:
18 / f( ) = = 4 4/ whenever 0. Now, notice that for large values of (in either the negative or positive sense) the values 3/ and -4/ will be very close to zero. Consequently, / f( ) = = as ± 4 4/ This process can always be used to observe a horizontal asymptote of a rational function. 3 + Eample: Consider the rational function r ( ) =. Notice that the denominator is 3 zero at = 3, and the numerator is not zero at = 3. Consequently, r has a vertical asymptote at = 3. Also, the degree of the numerator is the same as the degree of the denominator (namely ), so the leading coefficients tell us there is a horizontal asymptote at y = 3/ = 3. Notice that we can also see this by performing a calculation similar to the one shown in the remark above, since whenever 0. Consequently, / r ( ) = = 3 3/ / 3 r ( ) = = = 3 as ± 3 3/ Before we give the graph, we note that the y intercept occurs where = 0, and 30 ( ) + r(0) = = / 3. The intercept occurs where r() = 0, and 0 3 Finally, ( ) 3+ = 0 = /3 3 + r ( ) = 3 = = =
19 So the graph of r is the result of scaling g ( ) = by a factor of 4, shifting the result 3 units in the direction, and shifting 3 units in the y direction. The graph is shown below along with the vertical and horizontal asymptotes. Eercises:. Give the vertical and horizontal asymptotes of the rational function f( ) = Give the vertical and horizontal asymptotes of the rational function g ( ) = Give the vertical and horizontal asymptotes of the rational function 3 + h ( ) = Give the vertical and horizontal asymptotes of the rational function + R ( ) = Give the domains of the functions in problems through 4, and give the and y intercepts of their graphs Graph the function h ( ) = along with its vertical and horizontal asymptotes, and describe the behavior of h near the vertical asymptote. Describe how the graph of this function can be obtained from the graph of g ( ) = through a sequence of reflections, scalings and/or shifts Graph the function g ( ) = along with its vertical and horizontal 6 asymptotes, and describe the behavior of g near the vertical asymptote. Describe
20 how the graph of this function can be obtained from the graph of R ( ) = through a sequence of reflections, scalings and/or shifts Graph the function f( ) = along with its vertical and horizontal + asymptotes, and describe the behavior of f near the vertical asymptote. Describe how the graph of this function can be obtained from the graph of g ( ) = through a sequence of reflections, scalings and/or shifts. D. Eponential Functions Functions of the form f ( ) = a with a > 0 are called eponential functions. Notice that this function is very boring if a =. The behavior is very different depending upon whether 0 < a < or a >. We illustrate this by considering the cases a = ½ and a = below. Eample: Consider the function Notice that f( ) =. This function is defined for all values of. f( ) = = When is a large positive number, is a very large positive number. So, f() will be positive and very close to 0 when is a large positive number. In fact, we can make f() smaller than any given positive value by choosing positive and sufficiently large. We write this symbolically as f( ) 0 as Consequently, f has a horizontal asymptote at y = 0. On the other hand, when is negative and large in the negative sense, will be a very small positive number. So, f() will be a very large positive number. Finally, f can never be zero, so the graph of f does not have an intercept, and f(0) =, so the graph has a y intercept of. A table of values is given below along with a plot of f. Notice how the plot reflects the behavior mentioned above.
21 f() Eample: Consider the function f( ) =. This function is defined for all values of. Notice that when is a large positive number, is a very large positive number. In fact, we can make f() larger than any given positive value by choosing positive and sufficiently large. On the other hand, when is negative and large in the negative sense, will be a very small positive number. In fact, we can make f() smaller than any positive value by making sufficiently large in the negative sense. We say this symbolically as f( ) 0 as As a result, y = 0 is a horizontal asymptote for f. Finally, f can never be zero, so the graph of f does not have an intercept, and f(0) =, so the graph has a y intercept of. A table of values is given below along with a plot of f. Notice how the plot reflects the behavior mentioned above.
22 f() In general, the graphs of f ( ) = a for a > 0 and a 0 are similar to those for f( ) = and f( ) = given above. The plots are shown in the figures below. f ( ) = a for 0 < a < f ( ) = a for a > Eample: We can use the information above to graph the function f( ) =. The graph of f is the reflection of the graph of g= ( ) across the ais.
23 Eample: We can graph the function f( ) = by shifting the graph of the function g= ( ) by unit in the direction. A few points are plotted to illustrate the position of the graph. Eample: The graph of f( ) = can be found from the graph of g= ( ) 4. We first reflect the graph of g across the ais to obtain a graph of 4. Then we shift the resulting graph 0 units in the y direction. Notice that the horizontal asymptote for g at y = 0 gets shifted to the horizontal asymptote for f at y = 0. Reflection across the ais. Shifting in the y direction. The graph of f( ) =
24 + Eample: The graph of g ( ) = 6 can be given from the graph of g= ( ). We start by shifting the graph of g= ( ) by - unit in the direction. Then we shift the result -6 units in the y direction. Notice that this process shifts the horizontal asymptote for g= ( ) at y = 0 to the horizontal asymptote for g at y = -6. Shifting - units in the direction. Shifting -6 units in the y direction. The graph of + g ( ) = 6. Eercises:. Give the horizontal asymptote and the y intercept for the graph of g= ( ) 3.. Eplain why the graph of f( ) = is the same as the graph of f( ) = Give the horizontal asymptote and the y intercept for the graph of f( ) = Eplain how to use horizontal and vertical shifts to obtain the graph of g= ( ) 3 from the graph of f( ) =. 5. Eplain how to use horizontal and vertical shifts to obtain the graph of + 3 g ( ) = 7 from the graph of f( ) = Eplain how to use horizontal and vertical shifts to obtain the graph of g= ( ) 5 from the graph of ( ) 5 7. Graph g= ( ) Graph g ( ) = Graph g ( ) = +. bc b f =. Hint: a ( a ) c E. The Number e, Radioactive Decay, and Savings Accounts = when a > 0.
25 There is a positive number which is referred to as the natural eponent, and the number is so well known that it is given the special name e. The number e is an irrational number, so its eact value is not known. However, it can be approimated to many decimal places. In fact, rounded to 0 decimal places, e is given by e There are a number of websites which give e to more decimal places. For eample, you a quick search will return e rounded to 00 decimal places as e The reason that e is referred to as the natural eponent is because it shows up in so many important applications. Since e >, the graph of the eponential function f ( ) = e is very similar to those given in the section above. We show this graph below. The techniques from the previous section can also be used to graph f ( ) = e since e = e and 0 < /e <. Eample: Eperimental evidence has shown that radioactive substances decay in a such a way that the amount A(t) of the substance present at time t 0 (in years) is given by At () = Ce kt k( 0) 0, where k is a positive constant. Notice that A(0) = Ce = Ce = C since 0 e =. Consequently, the formula for A(t) can be rewritten in the form ( ) At () = A 0 e kt for t 0.
26 The formula for the radioactive decay of cobalt-60 has k = 0.3. We can use this information to determine the percentage of cobalt-60 remaining in an object after t years. The formula for A(t) above and the value for k gives 0.3t ( ) At () = A 0 e Consequently, the percentage of A(t) remaining (from an initial amount of A(0)) is given 0.3t by e. Some representative values are given in the table below. t (measured in years) Cobalt-60 percentage_remaining = e = 87.7% = 6.98% = 7.8% =.43% 0.3t A graph of the percentage function pt () 0.3t = e is shown below for 0 t 50. Eample: It can be shown that if a bank compounds interest continuously (at every instant) at an annual interest rate r (written in decimal form) then an initial investment of rt A 0 dollars will be worth At () = Ae 0 dollars t years later. If A 0 = 00, then the table below shows the resulting growth of the investment of $00 for a variety of interest rates. r =.03 r =.05 r =.07 r =. t = $03.05 $05.3 $07.5 $0.5 t = 5 $6.8 $8.40 $4.9 $64.87 t = 0 $34.99 $64.87 $0.38 $7.83 t = 0 $8. $7.83 $405.5 $738.9
27 Eercises: t. Graph the function pt ( ) = e+.. Graph the function gt ( ) = e t +. 0.t 3. Graph the function f ( t) = 00e. 4. A certain radioactive substance has a k value given by k = Create a table showing the percentage decrease in the substance for t =, 0, 50, 00, 500 years. 5. A certain radioactive substance has a k value given by k = Create a table showing the percentage decrease in the substance for t =, 0, 50, 00, 500 years. 6. A certain radioactive substance has a k value given by k = Estimate the ½ life of this radioactive substance. That is, estimate the time that must pass before 50% of the substance decays. 7. $000 is invested in an account that compounds interest continuously at a rate of 8%. How much money will be in the account after 0 years? 8. Your great great grandfather started a savings account 00 years ago with $00. Give an estimate for the amount of money currently in the account assuming the annual interest rate on the account has averaged 7% compounded continuously.
POLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationThe numerical values that you find are called the solutions of the equation.
Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationMidterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013
Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationNegative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2
4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationMAT12X Intermediate Algebra
MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions
More informationPower functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd
5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationTHE POWER RULES. Raising an Exponential Expression to a Power
8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More informationLINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,
LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationExponential Functions, Logarithms, and e
chapter 3 Starry Night, painted by Vincent Van Gogh in 889. The brightness of a star as seen from Earth is measured using a logarithmic scale. Eponential Functions, Logarithms, and e This chapter focuses
More informationis the degree of the polynomial and is the leading coefficient.
Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationFunctions: Piecewise, Even and Odd.
Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.
More informationZero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m
0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have
More informationAlgebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED
Algebra Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED. Graph eponential functions. (Sections 7., 7.) Worksheet 6. Solve eponential growth and eponential decay problems. (Sections 7., 7.) Worksheet 8.
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationGraphing Rational Functions
Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function
More informationSection 3-3 Approximating Real Zeros of Polynomials
- Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationExponential, Logistic, and Logarithmic Functions
5144_Demana_Ch03pp275-348 1/13/06 12:19 PM Page 275 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic
More informationLogarithmic and Exponential Equations
11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationSection 3-7. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative
202 Chapter 3 The Derivative Section 3-7 Marginal Analysis in Business and Economics Marginal Cost, Revenue, and Profit Application Marginal Average Cost, Revenue, and Profit Marginal Cost, Revenue, and
More information2-5 Rational Functions
-5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationSubstitute 4 for x in the function, Simplify.
Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationMULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.
1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationExponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014
Eponential Functions Eponential Functions and Their Graphs Precalculus.1 Eample 1 Use a calculator to evaluate each function at the indicated value of. a) f ( ) 8 = Eample In the same coordinate place,
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More information9 Exponential Models CHAPTER. Chapter Outline. www.ck12.org Chapter 9. Exponential Models
www.ck12.org Chapter 9. Eponential Models CHAPTER 9 Eponential Models Chapter Outline 9.1 EXPONENTIAL GROWTH 9.2 EXPONENTIAL DECAY 9.3 REVISITING RATE OF CHANGE 9.4 A QUICK REVIEW OF LOGARITHMS 9.5 USING
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationChapter 4: Exponential and Logarithmic Functions
Chapter 4: Eponential and Logarithmic Functions Section 4.1 Eponential Functions... 15 Section 4. Graphs of Eponential Functions... 3 Section 4.3 Logarithmic Functions... 4 Section 4.4 Logarithmic Properties...
More informationMPE Review Section III: Logarithmic & Exponential Functions
MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure
More informationPartial Fractions Decomposition
Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational
More information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More informationhttp://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304
MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationMSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationGraphing calculators Transparencies (optional)
What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used
More informationAlgebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationMath 131 College Algebra Fall 2015
Math 131 College Algebra Fall 2015 Instructor's Name: Office Location: Office Hours: Office Phone: E-mail: Course Description This course has a minimal review of algebraic skills followed by a study of
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction
More information4.6 Exponential and Logarithmic Equations (Part I)
4.6 Eponential and Logarithmic Equations (Part I) In this section you will learn to: solve eponential equations using like ases solve eponential equations using logarithms solve logarithmic equations using
More informationPolynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI-8 can be used
More informationPearson Algebra 1 Common Core 2015
A Correlation of Pearson Algebra 1 Common Core 2015 To the Common Core State Standards for Mathematics Traditional Pathways, Algebra 1 High School Copyright 2015 Pearson Education, Inc. or its affiliate(s).
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationChapter 7 - Roots, Radicals, and Complex Numbers
Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section
ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by
More information