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2011-2012 CP Precalculus Final Exam Topics Delaware Military Academy Chapter P & Chapter 1 NOT included on Final Examination The Final Exam will cover information from Chapters 2, 3, 4, and 10. Chapter 2 Section 2.1 Quadratic Functions Sketch the graph of a quadratic function Identify the axis of symmetry, vertex, and concavity Show familiarity with standard and vertex form of a quadratic Section 2.2 Polynomials of Higher Degree Identify the equations, degrees, and graphs of linear, quadratic, cubic, quartic, and quintic functions Complete a sign chart to sketch a polynomial written in factored form Section 2.4 Complex Numbers Apply operations on complex numbers Show understanding of the conjugate of a complex number Section 2.6 Rational Functions and their Asymptotes Understand and sketch transformations of the graph of y=1/x Given the graph of a rational function, describe the behavior of the graph Interpret the equation to find the vertical/horizontal asymptotes Discuss the domain and range of a rational function Chapter 3 Section 3.1 Exponential Functions and Their Graphs Sketch the graph of exponential functions, Identify the (horizontal) asymptote Use the formulas for Compound Interest [formulas given on final exam]. Section 3.2 Logarithmic Function and Their Graphs Evaulate a Logarithm without a calculator Converting between exponential and logarithmic form Sketch the graph of a logarithmic function Section 3.3 Properties of Logarithms Use the properties of logarithms to condense or expand a logarithmic expression Section 3.4 Solving exponential and Logarithmic Equations Solve simple or complex equations containing exponential or logarithmic functions Chapter 4 4.1 Radian and Degree measure Converting between radians and degrees Sketch and find angles that are coterminal 4.2 Trigonometric Functions: The Unit Circle Show your knowledge of all the degree, radian, and coordinates around the unit circle Show understanding of the ratios of sine, cosine, tangent, cosecant, secant, and cotangent Given one of the 6 trig functions, find one of the remaining 5 trig functions Evaluate any of the 6 trig functions for increments of 30, 60, 45, and 90 degrees around the unit circle. 4.3 Trigonometric Functions Use inverse trig functions to find a missing angle of a right triangle Use one of the 6 trig functions to find the missing side of a triangle. 4.5 Sketching Sine and Cosine waves Given an equation, sketch the sine or cosine wave Identify the amplitude and period Chapter 10 10.1 Circles Identifying the center and radius from a circle equation Given the equation, sketch the circle 10.2 Ellipses Identifying the major axis, minor axis, foci, vertices, co-vertices, and center of an ellipse Give the equation, sketch the ellipse 10.3 Hyperbolas Given the equation, sketch the hyperbola Identify the center and vertices FINAL EXAMINATION REVIEW QUESTIONS Section 2.1 Review Questions 1.) Locate the axis of symmetry of the quadratic f ( x) x 2 4 x 10 . Then find the vertex. 2.) Write the equation of a quadratic that is concave down. 3.) Rewrite the quadratic in standard from f ( x) 4( x 3) 2 5 . 4.) Identify the vertex of the following quadratic f ( x) 4( x 3) 2 5 . Section 2.2 Review Questions 5.) Write an example equation and graph of a linear, quadratic, cubic, and quartic function. 6.) Complete the sign chart for the following polynomial. f ( x) x( x 3)( x 1) x -∞<x<-3 -3 -3<x<0 0 0<x<1 1 1<x<∞ f(x) 7.) Sketch the polynomial that corresponds to the following sign chart. x -∞<x<-4 -4 -4<x<-1 -1 -1<x<1 f(x) 1 1<x<∞ Section 2.4 Review Questions 8.) Write the imaginary number, i, as a radical. 9.) Complete the operation: (2+2i)-(3+4i) 10.) Complete the operation: (4i)(-2i) 11.) What is the conjugate of (2+3i)? 12.) Rationalize the denominator by multiplying by the conjugate of the denominator: 3 (2 + i) Section 2.6 Review Questions 1 1 1 1 13.) Sketch the graphs of y , y 1 , y , and y . List the asymptotes x x -1 x 1 x of each 4x 2 - 16 14.) Find the vertical and horizontal asymptote of the rational function. y 2 x 9 x 15.) Write the end behavior model of the rational function y 2 . Explain why this x 9 graph has no horizontal asymptote. 16.) Describe the behavior of the rational function graphed below. Section 3.1 Review Questions 17.) Sketch the graph of y 3 x , y 3( x 2) , and y 3 x 2 . Identify the horizontal asymptote of each. 18.) Suppose that $500 is compounded continuously for 5 years in an account with an interest rate of 2%. How much money will be left in the account after 5 years? 10 years? 19.) Suppose that $500 is compounded weekly for 5 years in an account with an interest rate of 2%. How much money will be left in the account after 5 years? What if the money is compounded monthly? Section 3.2 Review Questions 20.) Evaluate each logarithm without using a calculator. 1 log e 1 log 4 16 log 3 9 log 10 10 21.) Write the equation in logarithmic form. 2 x 256 22.) Sketch the graphs of y log 2 x , y log 2 ( x 2) , y log 2 ( x) 2 Label and identify the asymptotes of each Section 3.3 Review Questions 23.) Use the properties of logarithms to expand the following term into the xy sum/difference of logarithms. ln( 3 ) z 24.) Use the properties of logarithms to condense the following expression into a single term logarithm. log 4 ( x) 2 log 4 (2 y) 3 log 4 ( x) Section 3.4 Review Questions 25.) Solve the following logarithmic equations. log x 16 4 2 log 2 ( x 4) 2 3 ln( x) 5 26.) Solve the following exponential equations. 4 ( 2 x 1) 64 10 x .001 Section 4.1 Review Questions 27.) Convert 410 to radians 5 28.) Convert into degrees. 8 29.) Find two angles that are coterminal to each other. Section 4.2 Review Questions 30.) Fill in the following portion of the unit circle. 31.) Evaluate the following trig expressions. tan( 0) sin( ) sec( ) 3 4 32.) If sin(x)=3/4, then cot(x)=____. Section 4.3 Review Questions 33.) Use inverse trig to find the missing angle. 34.) Find the identified missing side length of the each triangle. cos( ) 2 Section 4.5 Review Questions 35.) Sketch each function. Label the axii. y sin( x) , y cos(x) , and y 5 cos( x) . 2 36.) Identify the period and amplitude of the following: y 5 cos(2 x) , y sin( 1 x) 8 Section 10.1 Review Questions 37.) Identify the center and radius of the following. ( x 3) 2 y 2 1 38.) Sketch the following circle. y x 2 9 Section 10.2 Review Questions 39.) Sketch the following ellipse. Label the minor axis, major axis, foci, center, vertices, ( x 3) 2 and co-vertices of the ellipse. y2 1 25 Section 10.3 Review Questions 40.) Sketch the following hyperbola. Label the center and vertices. ( x 1) 2 y 2 1 9 4