Mth 138 College Algebra Review Guide for Exam III Thomas W. Judso Stephe F. Austi State Uiversity Sprig 2018 Exam III Details Exam III will be o Thursday, April 19 ad will cover material up to Chapter 6 with emphasis o the material covered sice the last exam. Calculators will be allowed o this exam. If you qualify for a accommodatio, please make arragemets with Disability Services as soo as possible. If you wait util the day of the exam to make arragemets, we will ot be able to meet your accommodatio eeds. Studyig ad Reviewig You should try workig some of the problems i the review sectios of Chapters 4, 5, ad 6. Be sure to take advatage of the office hours, learig teams, ad the AARC. Topics for Exam III Be sure to familiarize yourself with topics covered i the first part of the course (material from Exams I ad II). To fid a expoetial fuctio f(x) = ab x through two poits: 1
1. Use the coordiates of the poits to write two equatios i a ad b. 2. Solve for b. 3. Substitute b ito either equatio ad solve for a. Doublig Time: If D is the doublig time for a expoetial fuctio P (t), the P (t) = P 0 2 t/d. Half-Life: If H is the half-live for a expoetial fuctio Q(t), the Q(t) = Q( 0 (0.5) t/h. Future Value of a Auity: If you make paymets per year for t years ito a auity that pays iterest rate r compouded times per year, the future value, F V of the auity is [ ( P 1 + r ) ] t 1 F V = r, where each paymet is P dollars. Preset Value of a Auity: If you wish to receive paymets per year for t years from a fud that ears iterest rate r compouded times per year, the preset value, P V of the auity is [ ( P 1 1 + r ) ] t 1 P V = r, where each paymet is P dollars. Suppose g is the iverse fuctio for f. The g(b) = a if ad oly if f(a) = b. We usually write f 1 (x) for the iverse fuctio of f(x). Horizotal Lie Test: If o horizotal lie itersects the graph of a fuctio more tha oce, the its iverse is also a fuctio. Such a fuctio is called a oe-to-oe fuctio. The iverse of f is a fuctio if ad oly if f is oe-to-oe. Fuctios ad Iverse Fuctios: Suppose that f 1 is the iverse fuctio for f. The f 1 (f(x)) = x ad f(f 1 (y)) = y as log as x is i the domai of f ad y is i the domai of y. 2
The Fudametal Priciple of Logarithms: For b > 0, b 1, ad x > 0, y = log b x if ad oly if x = b y. The Logarithmic Fuctio: The logarithmic fuctio base b, g(x) = log b x, is the iverse of the expoetial fuctio of the same base, f(x) = b x.. 1. Domai: all positive real umbers 2. Rage: all real umbers 3. x-itercept is (1, 0) 4. There is o y-itercept 5. Vertical asymptote at x = 0 6. The graphs of y = log b x ad y = b x are symmetric about the lie y = x. Expoetial ad Logarithmic Fuctios: Because f(x) = b x ad g(x) = log b x are iverse fuctios for b 1, ad x > 0, for x > 0. log b b x = x ad b log b x = x Steps for Solvig Logarithmic Equatios: 1. Use the properties of logarithms to combie all logs ito oe log. 2. Isolate the log o oe side of the equatio. 3. Covert the equatio to expoetial form. 4. Solve for the variable. 5. Check for extraeous solutios. The atural base is a irratioal umber called e, where e 2.71828182845 The atural expoetial fuctio is the fuctio f(x) = e x. The atural log fuctio is the fuctio g(x) = l x = log e x. Coversio Formulas for Natural Logs: Properties of Natural Logarithms. l(xy) = l x + l y y = l x if ad oly if e y = x 3
( ) x l = l x l y y l x p = p l x We use the atural logarithm to solve expoetial equatios with base e. Expoetial Growth ad Decay: The fuctio P (t) = P 0 e kt describes expoetial growth if k > 0, ad expoetial decay if k < 0. Cotiuous compoudig: The amout accumulated i a accout after t years at iterest rate r compouded cotiuously is give by where P is the pricipal ivested. A(t) = P e rt Quadratic Fuctios: A quadratic fuctio is oe that ca be writte i the form ax 2 + bx + c, where a, b, ad c are costats with a 0. Zero-Factor Priciple: ab = 0 if ad oly if a = 0 or b = 0. x-itercepts of a Graph: The x-itercepts of the graph y = f(x) are the solutios of the equatio f(x) = 0. Stadard Form of the Quadratic Equatio: ax 2 + bx + c = 0 Factored Form of the Quadratic Equatio: a(x r 1 )(x r 2 ) = 0 To Solve a Quadratic Equatio by Factorig: 1. Write the equatio i stadard form. 2. Factor the left side of the equatio. 3. Apply the zero-factor priciple: Set each factor equal to zero. 4. Solve each equatio. There are two solutios (which may be equal). Every quadratic equatio has two solutios, which may be the same. 4
The value of the costat a i the factored form of a quadratic equatio does ot affect the solutios. Each solutio of a quadratic equatio correspods to a factor i the factored form. A equatio is called quadratic i form if we ca use a substitutio to write it as au 2 + bu + c = 0, where u stads for a algebraic expressio. The Quadratic Formula The solutios of the equatios ax 2 + bx + c = 0, with a 0 are x = b ± b 2 4ac 2a The graph of a quadratic fuctio f(x) = ax 2 + bx + c is called a parabola. The values of the costats a, b, ad c determie the locatio ad orietatio of the parabola. The Discrimiat: The discrimiat, D = b 2 4ac, of a quadratic equatio determies the umber of x-itercepts of a quadratic fuctio. 1. If D > 0, there are two uequal solutios. 2. If D = 0, there is oe real solutio of multiplicity two. 3. If D < 0, there are two complex solutios. To Graph the Quadratic Fuctio: 1. Determie whether the parabola opes upward (if a > 0) or dowward (if a < 0). 2. Locate the vertex of the parabola. (a) The x-coordiate of the vertex is x v = b 2a. (b) Fid the y-coordiate of the vertex by substitutig x v ito the equatio of the parabola. 3. Locate the x-itercepts (if ay) by settig y = 0 ad solvig for x. 4. Locate the y-itercept by evaluatig y for x = 0. 5. Locate the poit symmetric to the y-itercept across the axis of symmetry. To uderstad ad be able to apply quadratic models.. Vertex Form for a Quadratic Fuctio: A quadratic fuctio y = ax 2 + bx + c, with a 0, ca be writte i the vertex form where the vertex of the graph is (x v, y v ). y = a(x x v ) 2 + y v, 5