Brooklyn College Department of Mathematics. Precalculus. Preparatory Workbook. Spring Sandra Kingan

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1 Brooklyn College Department of Mathematics Precalculus Preparatory Workbook Spring 0 Sandra Kingan Supported by the CUNY Office of Academic Affairs through funding for the Gap Project

2 CONTENTS. Review of Pre-Algebra..... Fractions..... Irrationals..... Integer exponents..... Radicals..... Properties of real numbers.... Algebraic Thinking..... Algebraic expressions..... Linear equations..... Visualizing inequalities..... Solving inequalities..... Absolute value equations and inequalities Vizualization in Algebra..... Graphs of straight lines..... Finding the equation of a line..... Functions and graphs... ii

3 . REVIEW OF PRE-ALGEBRA.. FRACTIONS ) Simplify the fractions by writing them in lowest terms ) Express each fraction as a mixed number ) Express each mixed number as a fraction. 0 0

4 ) Add/subtract fractions with like denominators. Leave your answer in lowest form ) Add/subtract fractions with unlike denominators. Leave your answer in lowest for ) Multiply the fractions. Leave your answer in lowest form. ( )( ) 0 0 ( )( ) 8 8 ( )( ) 8 ) Multiply the fractions. Leave your answer in lowest form. 8 0

5 .. IRRATIONALS ) Mark the specified decimals on the number line. a) Mark every tenth point between 0 and. 0 b) Mark every tenth point between 0 and ) Compare the decimals. Write the symbols <, >, or ) Write each repeating decimal using the bar notation. Decimal Bar Notation

6 ) Write each fraction as a decimal using your calculator. List as many decimal places as needed to recognize the pattern ) Write each fraction as a decimal by guessing the pattern. 0 00, , , , ,000

7 0 00, , , ,000 0,000 ) Write each fraction as a decimal by guessing the pattern. 0 00,000 0, ,000 0, ,000 0, ,000 0, ,000 0, ,000 0, , , ,000 0,000

8 ) What is the error when we approximate π by the fraction. 8) Draw a number line and mark the integers 0 to 0. Then mark π and e. ) Fill in the table below using your calculator. Which square-roots are irrational? Number 8 0 Square root (at least up to 8 decimal places) 0) Write each decimal as a fraction in lowest form

9 .. INTEGER EXPONENTS ) Fill in the table. Exponential Notation Base Exponent Words raised to the raised to ) Write each multiplication expression as an exponential notation. a) b) c) d) e) a a a a f) g) h) y y y y y y y y y y y y i) x x x x x x x x j) ( )( )( )( ) k) ( )( )( )( )( ) l) ( x )( x)( x) ) Write each exponent as a multiplication expression and simplify, if possible. a) b) ( ) ( )( )( )( ) c) d) ( ) e) ( ) f) g) h) 0 i) x j) ( x ) k) y

10 ) Fill in the blanks ) Write each negative exponent as a positive exponent. j) y a) k) ( a ) b) ( ) ( ) l) ( ) c) d) ( ) e) ( ) f) g) h) 0 i) x m) ( ) n) ( ) o) ( ) p) ( ) 8

11 ) Write the following numbers in scientific notation , , , ,000, ,000, ,000, ,000,000, ,000,000, ,000,000, ) Write each number in scientific notation. 0,000,000,000,000,0,000, ) Write each number in standard form ) Large and small number are everywhere in nature. Complete the following table. Physics constant Number Scientific Notation Speed of light in a vacuum,,8 meters/second Diameter of an atom 0 meters Diameter of the nucleus of an atom 0 meters Atomic mass Approximate age of the universe. billion years.0,8,8 0 Number of stars 0

12 m n m+ n 0) Simplify using the exponential propertiesb b b, m n n n n b, ( ) a b a) ( )( ) k) b) x y l) c) x y b b m n ab and a b n a b n n d) x x e) x y x y f) x y x y g) x y x h) i) m) ( ) n) o) (xy) x p) y xy q) a x j) x ) (Number sense) If the beginning of the line is 0 and the end is billion, where would you mark million? 0

13 .. RADICALS ) Find the values of the following radicals without using a calculator: ) Find the values of the following radicals. Use your calculator.. π ) Simplify using radical properties. Do not use your calculator. a) d) 8 b) 8 e) 0 c) 8 f) 00

14 .. PROPERTIES OF REAL NUMBERS ) What are the following properties of real numbers. Think about the most elegant and concise way of expressing these properties a) The commutative property of addition b) The commutative property of multiplication c) The associative property of addition d) The associate property of multiplication e) The distributive property f) The additive identity property g) The multiplicative identity property h) The additive inverse property i) The multiplicative inverse property ) Identify the property shown in each equation: a) + + f) + 0 b) g) c) + ( + 8) (+ ) + 8 h) + ( ) 0 d) ( 8) ( ) 8 e) 8 ( + ) 8() + 8() i) j) () + () ( + )

15 ) Rewrite each expression using the distributive property. No need to simplify. a) ( + 8) () + (8) b) 8 ( 0) c) ( ) d) ( + ) e) ( ) f) ( ) g) ( ) h) ( + ( )) i) ( + ) j) ( + ) k) (8 ) ) Answer the following questions. a) What is the additive identity for real numbers? b) What is the multiplicative identity for real numbers? c) What is the additive inverse of -? d) What is the multiplicative inverse of? e) What is the multiplicative inverse of f) What is the additive inverse of? g) What is the additive inverse of?? ) Give examples to show that the commutative and associative properties do not hold for subtraction and division.

16 . ALGEBRAIC THINKING.. ALGEBRAIC EXPRESSIONS ) Evaluate the algebraic expression x for the different values of x given below: a) x b) x c) x 0 y ) Evaluate the algebraic expression 8 for the different values of y given below: a) y b) y c) y 0 ) Express each phrase as an algebraic expression. a) A number x times b) 8 plus a number x c) 8 times a number x d) A number y minus e) A number z increased by f) A number z decreased by g) Subtract a number x from 0 h) Subtract 0 from a number x i) The product of 0 and a number z j) A number divided by k) divided by a number x l) Add to a number x times m) Add a number x to 0, then divide by 0 n) Subtract from the product of x and

17 ) Expand each expression using the distributive property: a) ( x + 8) f) ( x + y + ) b) 8( x 0) c) ( x) d) ( x + ) e) ( x + y) g) ( x + y z + w) h) ( x y + z ) i) ( x + ) j) ( x + ) ) Simplify the algebraic expressions. a) x + x i) ( x + ) ( x ) b) x y + x j) x + y + 8( x + y) c) ( x + ) + k) x + y 8( x y) d) x + x + e) ( x + ) + f) ( x ) + l) xy + ( xy + z) m) ( x + ) + n) (x + ) (x ) g) x x + 0) ( x + ) + (x + ) h) (x ) p) ( x + ) + (x + )

18 .. LINEAR EQUATIONS ) Solve the linear equations. a) x + b) x c) x d) x + ) Solve the linear equations. a) x 0 x b) c) x d) x ) Solve the linear equations. a) x + b) x + c) x d) x ) Solve the linear equations. x a) + b) x x + c) d) x + 8 ) Solve the linear equations. a) x + x + 0 b) 0 x + x + x c) x + x d) x + x e) + x 0 f) g) x x + x + x ) Solve the linear equations. a) x + ( x + ) b) ( x + ) ( x + ) c) x + ( x + ) x + d) ( x + ) x +

19 .. VISUALIZING INEQUALITIES ) Draw each inequality on a number line and write it as an interval (as shown): a) < x < (,) h) < x < b) x [,] i) x c) x < [,) j) x < d) < x (,] k) < x 0 e) x > (, ) l) x > f) x m) x (,] g) < x < n) x (, )

20 8 ) Draw each interval on a number line and write it as an inequality. a) (, ) e) (,] b) (,) f) (, ) c) [,] g) [, ) d) [,) h) [, ) ) Jane labels the interval below as (, ). Is she right? Explain. ) Joshua labels the interval below as [,]. Is he right? Explain. 8

21 .. SOLVING INEQUALITIES ) Solve the inequalities. Draw your answer on the number line and write it in interval form. a) x 0 b) x 0 c) x + > d) x 0 e) x f) x 0 g) x 0 h) x x i) ( x + ) > x + ) Solve the compound inequalities. a) x + < b) < x + < c) < x + <

22 .. ABSOLUTE VALUE EQUATIONS AND INEQUALITIES ) Solve the following absolute value equations. a) x b) x + c) x 8 d) x + e) x f) x + x g) x h) x x + i) x + x 0

23 ) Solve the following absolute value inequalities. a) x b) x > c) x d) x e) x < f) x + > 8

24 . VIZUALIZATION IN ALGEBRA.. GRAPHS OF STRAIGHT LINES ) Draw a coordinate system. Plot the following points and state which quadrant they lie in. A (,) B (,) C (, ) D (, ) E (,) F (,) G (,) H (, ) I (, ) J (,0) K (,0) L (0,) M ( 0, ) N (0,) O ( 0, ) P (,0) ) Find the distance between the two points. Also find the midpoint. a) (,) and (, ) b) (,) and (,) c) (, ) and (,0) d) ( 0, ) and (,0) ) Graph the following linear equations by plotting some points. a) y x c) y x b) y x d) y x +

25 ) Graph the following linear equations by finding the x-intercept and y-intercept. a) y x + b) y x + c) y x + ) Graph the following horizontal and vertical linear equations. a) y b) y c) x d) x ) Find the slope (if it exists) of the line joining the two points. a) (,) and (,) c) (,) and (, ) b) (,) and (,8) d) (,) and (, ) ) When the equation of the line is given in the slope-intercept form y mx + b we can read off information as follows: slope is m and y-intercept is b. Practice reading this information from the following equations. a) y x + d) y x b) y x + c) y x e) y x +

26 .. FINDING THE EQUATION OF A LINE ) Find the equation of the line passing through the point P and with slope m. a) P (,) and m b) P (,) and m c) P (, ) and m d) P (, ) and m 0 e) P (, ) and slope is undefined ) Find the equation of the line passing through the points P and Q. a) P (, ) and Q (,) b) P (0,) and Q (,) ) Determine if the lines are parallel, perpendicular, or neither a) y x + and y x + b) y x + and y x + c) y x + and y x + d) y x + and y x + e) y x + and y x + f) x + y + 0 and x + y g) y and x 0 h) y and x ) Find the equation of the line that is parallel to and the line that is perpendicular to the given line and passing through the given point. a) y x + and P (, ) b) y x + and P (, ) c) y and P (, )

27 .. FUNCTIONS AND GRAPHS ) Determine whether the correspondence is a function. a) b) c) d) e) f) ) Determine whether the relation is a function. Identify the domain and range. a) {(,0),(,),(,0 )} b) {(,),(,),(, )} c) {(,),(,),(,),(0,)} d) {(,),(,),(,),(, )} e) {(,),(0,),(,),(,),(, )} f) {(,0),(, ),(0,0),(, ),(, )} ) Find the specified function values: a) f ( x) x f () f ( ) f (0) f (a) b) g ( x) x g () g ( ) g (0) g (a) c) h ( x) x h () h ( ) h (0) h (a) ) Find the specified images and preimages: a) f ( x) x image of is pre-image of 0 is b) f ( x) x image of is pre-image of 0 is c) f ( x) x image of is pre-image of 0 is

28 ) Use the vertical line test to determine if y is a function of x. If it is a function use the horizontal line test to determine if it is a one-to-one function. y x y x x y x + y x xy x y

29 ) Find the domain and range of the following functions. Where is the function increasing, decreasing or constant? (Note that the ends go off to infinity even though there are no arrows. This is because graphing software does not put arrows.) f ( x) x f ( x) x + 0 f ( x) x + x + x + f ( x) x x f ( x) x f ( x) x f ( x) x x f ( x) x

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).