Grade 11 Functions (MCR3U) Unit 1 Algebraic & Quadratic Functions

Transcription

1 Grade Functions (MCRU) Unit Algebraic & Quadratic Functions Topics Homework Tet book Worksheet Da Order of Operations Order of Operations #- Da Eponent Laws Review Eponent Laws Review (All) Da Review of Algebra Review of Algebra (All) Da Review of Factoring Review of Factoring (All) Da Review of Factoring Review of Factoring (All) Da Functions, Domain and Range Functions, Domain and Range.: P. #-, 7, -8 b Graph (All) b Graph.: P. #,, Factoring Mied Review Da 7 Radical Sstem.: P. 9 #-8 (Ever other), 9,, a Radical Sstems (All) Da 8 Comple Number Sstem Comple Numbers Sstem (Optional) #-9 Da 9 Solving Quadratic Equations b Solving Quadratic Equations.: P. 9 #, Factoring (B factoring) Da 0 Review of Quadratic Functions Review Quadratics (All.: P. #- (All tpes) tpes) Odd # Da Review of Quadratic Functions Review Quadratics (All.: P. 9 #, -8,b, (All tpes) tpes) Even # Da Graphing of Quadratic Functions.: P. 7 #,,-8, Da Quadratic (Ma & Min) Quadratic (Ma & Min).: P. #, 9,-- Problems Problems (All) Da Quadratic (Ma & Min) Quadratic (Ma & Min).:, Problems (Da ) Problems (Da ) (All) Da Quadratic Word Problems Quadratic Word Problems (Roots) (Roots) (All) Da Da 7 Da 8 Intersection of Lines and Nonlinear Relations Review Test P. 8 #,,bc,0,, P. 70. &. for more problems Chapter Tetbook Review Intersection of Linear and Non-Linear Relations (All)

2 Grade Mathematics Page of Order of Operations The mathematical operations (addition, subtraction, multiplication, division, and eponentiation) are done in a specific order. The rules for this order are: The Acronm for the order is BEDMAS. All calculations must be done from left to right.. Do the calculations within the brackets (parenthesis). When ou have more than one set of brackets, do the inner brackets first.. Eponents (or radicals) must be done net.. Then, do multiplications/divisions in the order the appear from left to right.. Then, do additions/subtractions in the order the appear from left to right. Eample : Evaluate a) 0 b) 0 c) 8 d) 8 e) 8 f) g) h)

3 Grade Mathematics Page of Order of Operations. Evaluate a. e. b. f. 7 c. g. 8 h. d. 7 0 i. j. 7. Evaluate 9 a. 0 b. c. d. 9 8 e. 7 f. 7 0 g. h i. 9 7 j Evaluate a. b. 8 c. 9 d. 8 7 e. 7 f. 8 9 g. 0 h i. 8 j.

4 Grade Mathematics Page of Order of Operations. Evaluate a. 0 b. c. d e. f. g. h i. j. 8. Evaluate the epression a. b. c. d. 8 7 ========================================================================== Answers a. b. c. 0 d. e. f. g. 0 h. 0 i. 0 j. 9 a. b. c. 9 d. e. f. g. h. 8 i. j. 7 8 a. b. 9 c. d. e. f. g. 7 h. i. j. 7 a. b. 7 0 c. d. 0 7 e. 9 f. g. h. 9 7 i. j. 8 a. b. 8 c. d. 7

5 Grade Mathematics Page of Eponent Laws Review Review of the eponent laws If m and n are an real numbers, then m n mn ) a a a m a mn ) a ( a 0) n ) ) a m n ( a ) a n mn ( abc) a n n a a ) ( b 0) n b b n b n c n Negative Eponents ) 7) a m a b a m m b a m Zero Eponents 0 8) a ( a 0) ( a 0) ( m N) Eample : Evaluate a) b) c) ( ) d) ( ) e) f) g) Eample : Epress answers in positive eponent a) a a 7 7 b) a a c) a a d) 0 a a Eample : Epress answers in positive eponent a) ) (a b) ( a ) c) ) (ab d) a b e) ( a b ) Eample : Simplif 0a b c a) 7 a bc b) z z 7 z c) 7 z

6 Grade Mathematics Page of Eponent Laws Review Simplif each of the following: (Positive eponents). =. =. ( ) =. ( ) =. =. ( ) = 7. ( ) = 8. = ( ) 9. = 0. =. =. =. (9 ( )( ) =. )(0) 0 0 =. 0 =. ( ) 0 = 8 Epress as a single power: (Positive eponents) 7. a) a b) c) m d) s s e) t f) r r g) h) a a 8. a) 7 9 b) c) d) m m e) m m f) 7 9 a a g) h) 7 Simplif: (Positive eponents) 9 a) b) c) a a d) 7 7t t e) 7 c c f) m m g) 7 m m h) 8 r r 9 8t 0. a) 9t 7 r s b) r s 7m n c) 9m n d) e) r r f) a a g) 0 h) m m Simplif: (Positive eponents). a) (a ) b) ( ) c) (m ) d) ( 9 ). a) ) (m b) (t ) c) (c ) d) ( ) e) 7 ) (u f) 7 ( ) g) (ab) h) ( ) a i) b j) a k) b l) ab c

7 Grade Mathematics Page of Eponent Laws Review. a) ( a b )( a b ) b) ( ab )( a b ) c) ( z )( z ) d) ( )( ) e) ( m n ) ( mn ) f) ( mn ) ( mn ) 8 a b. a) a b m n d) 7 9 m n 8 b) e) 8 a b c) a b a b f) a b. a) ( ) b) ( 7 ) c) ( m n ) d) ) 8 (abc e) ( ) f) ( z ) Rewrite using positive eponents:. a) a b) c) m d) r e) m f) a b g) a b h) a b i) m j) r k) a b l) a b Rewrite using positive eponents, then evaluate: (Show one middle step.) 7. a) b) c) 9 d) e) 7 f) 0 g) h) i) Simplif. Give all answers with positive eponents onl: (Show at least one middle step.) 0a b 8. a) a b 7a b c d) 8 9a b c g) a b c j) 8 a b c 8m n q b) 7 m n q m n p e) m n p z h) z z k) z 0m n c) m n 8a b f) a b 9a b c i) a b c z l) z

8 Grade Mathematics Page of Eponent Laws Review 9. Simplif. Give all answers with positive eponents onl. (Show at least one middle step.) ( mn ) a) m n b) ( ) ( ) c) 7 ( s t) d) s t ( mn ) e) m n ( ab ) f) a b s t g) s ( t ) 0( a ) b h) a b ( ) z i) 7 ( z) Answers.. g. 9a. 0a. c h. a b. t b. m d. a b k. (m n q ) j) (m n q ) a. 8a. c. s r c. m 8 a. 8 a b c l. c. d. m n e. i f. m j. 0 r k. a. e. z b. 7 a d. mn d. b. 9 t c. 0 t e. e. c. a b a b b. f. 8 a. a b l. c d. c. z a b a. a ab 7a. f g. h. 7 i. n p e. m a f. 7 b b. ( ) z k) 8( ) z a b. 8 7 d. m 8c f. r f. m 9 a g. u e. 8 d. 9 b. c. m () l) ( ) 0. c. m d. s 0 e. t e. m m m g. 8m h. h. m e. 9 f. g. m 7 n f. c. d. r e. m b. c. 9 9 a 8a. 8 b 9 z 9 ac g. h. 7 i. j. b a bc b. m n k. f. a g. 8r a. a b. a b h. n z m n a. 0 m d. f. a b. f. r h. a i. b a b c g. j. 7 a b b. a b d. e. 7 0 c. m 9 n d. l. z c a b 9a. n h. ab b. 7 c. d. 7 s e. m f. 7b a 7t g. h. s i. a 8 b z 8 8m j. q n z k. l.

9 Grade Mathematics Page of Review of Algebra Polnomial: A mathematical algebraic epression involves with numbers, eponents and variables. Monomial: term polnomial Binomial: terms polnomial Trinomial: terms polnomial Like Terms: Polnomials with the same variable(s) and eponents Eponent Laws: a a m m ( a a m n n a ) n a a a mn mn mn a b m a b m n Steps are for our part marks!!! Show steps!!! Be careful to the brackets & signs!!! Eample : Grouping Like Terms Simplif the following a) ( 7 ) ( ) b) ( 7 ) ( ) c) ( ) d) ( ) ( 7) ( ) e) Some useful patterns Eample : Distributive Properties Simplif the following a) b) 7 7 c) e) f) d)

10 Grade Mathematics Page of Review of Algebra. Simplif a) 7a b c 9a b c b) p q r r p s p s r q c) z 7 z z d) 7a 7 a 9 e) a 7 7 a 7 9 a) From p p subtract p. b) Subtract the sum of and from the sum of,, and.. Epand and simplif. a) d) e) b) g g c). Epand and simplif. a) b) c) d) e) a c a 7c f. Epand and simplif. a) b) 7 c) 7 d) e) k k k k f) p p p. Epand and simplif. a) a b c b) z c) d) e) z z z f) g) Answers: a) a b 8c b) p r s c) z d) a 8 e) a) p p b) c) a) b) g 0g c) 7a d) e) 9 a) b) 8 c) 8 d) 9 e) a c 9ac f) 9 a) b) 8 c) 0 0 d) e) k 7k f) p p a) a b c ab bc ac b) 9 z z z c) 0 0 d) f) g) e) z 9z 9z 9

11 Grade Mathematics Page of Review of Factoring Topics: Common Factoring, Factoring b Grouping, Factoring Difference of Squares Eample : Great Common Factor (GCF) Find the GCF of the following, 9, Eample : Common Factors a) b) z Eample : Factoring Polnomials b Grouping a) a(a b) b(a b) b) 7 c) 7 d) k 8k k 0 Eample : Factoring Difference of Squares a) 9 b) 9 b a b a ba b c) 7 9 d) a 8b 8

12 Grade Mathematics Page of Review of Factoring Topics: Common Factoring, Factoring b Grouping, Factoring Difference of Squares Factor (highest common) each of the following:. a) b) c d c) 7a 7b d) e) a az f) a b g) b b h) a b. a) b) a ab c) d) e) a a f) a a a g) a b ab h). a) 0 z b) a a az c) d) 8 e) a a az f) z g) h) a a a. Factor (Two steps for some): a) a + a + b + b b) b + c + ab + ac c) d) + e) + f) +. Factor: (Two steps Common): a) ( ) + b) ( + ) + + c) (a + ) a a d) ( ) + e) (ab ) ab + f) ( a) + ( a). Factor into the product of two binomials. a) 9 b) b c) 9m d) 8f e) f) 9 g) 8 h) t i) m 8n j) k) 9a b l) b 00c m) ( ) z n) (a b) 8 o) ( m ) (m )

13 Grade Mathematics Page of Review of Factoring Answers: a) b) c d c) 7 a b d) e) a z f) a b g) b h) a b a) a a b a a a g) ab a b b) f) a) z b) a z e) a z f) z c) h) d) e) a a c) d) g) h) a a a a) a b b) b c a d) e) f) c) a) b) c) a a d) e) ab ab f) a a a) 7 7 b) b b c) m 8 m 8 d) 9 f 9 f e) f) 7 7 g) 9 9 h) t t i) m 9nm 9n j) 8 8 k) 7a b 7 a b l) b c b c m) z z n) a b 9a b 9 o) m m

14 Grade Mathematics Page of Review of Factoring Topics: Special Trinomial, Simple Trinomial, Non-Simple Trinomial Eample : Factoring Special Trinomials a) b) 0 0 a a ab b ab b a b a b Eample : Factoring Trinomials +b+c a) 7 b) 8 9 c) 7 0 d) a 0a b 0b Eample : Factoring Trinomials a +b+c a) 7 b) c) Eample : Combining Factoring Techniques a) z z 0 z b) z z c) a 9b c bc

15 Grade Mathematics Page of Review of Factoring Topics: Special Trinomial, Simple Trinomial, Non-Simple Trinomial Factor into the product of two binomials.. a) 8 b) c) a a d) p p e) 9 f) t 0t. a) 7 0 b) a a c) m 0m d) a 8a 7 e) 8 f) n n 0. a) 8 b) c c c) a 7a d) e) f) n 9n. a) b) 0 c) a a 7 d) n n 0 e) m m f) 8 7. a) 8 b) 9 c) d) p q 0 pq e) c d 9cd f) 7. a) 0 b) a 0a 0 c) 0n 0n 0 d) a a 0 e) f) 7a a g) h) a a a i) j) 8m 7 k) 0 l) 0 7. a) 7 b) a a c) d) k k e) s s f) a) 7 b) n n c) c c d) e) 7 f) a a 9. a) t 7t b) k k c) 8r r d) m m 0 e) 9 f) d d

16 Grade Mathematics Page of Review of Factoring 0. a) a 7a b) 0 c) m m d) k 9k 9 e) f) a a. a) b) m m c) s 0s d) 9 e) 7 f) 8a a. a) 0 b) k k c) g 7g d) 9p 7 p e) 8c 8c f). a) b) h h c) q q d) 0u 9u 0 e) 0m 7m f) 8c c. a) h 7h b) 0r r c) w w d) t 9t e) 0 7 f) 9a a. a) b) a a 0 c) 8a a 8 d) r r 0 e) 7 f) a a 0. a) b) a a 0a c) 8 d) 0m m 0m e) 9a 9a a f) ab 9ab a 7. a) 0 b) s s c) a 9a d) 8 e) 0a 9a f) 8. a) 7 0 b) 7 c) 8 d) 8 e) c cd d f) 0 z z

17 Grade Mathematics Page of Review of Factoring Answers a) b) c) a d) p e) f) t a) b) a a c) m m d) a 7a e) f) n n 0 a) b) c 8c c) a a d) e) f) n 7n a) 8 b) 0 c) a 9a 8 d) n 8n e) m 7m f) 8 a) b) 7 c) d)pq 7 pq e) cd cd f) a) b) a a c) 0n n d) a a e) f) 7a a g) h) a a a i) j) 8m m k) l) 7a) ( )( ) b) ( a )( a ) c) ( )( ) d) ( k )( k ) e) ( s )( s ) f) ( )( ) 8a) ( )( ) b) ( n )( n ) c) ( c )(7c ) d) ( )( ) e) ( )( 7) f) ( a ) 9a) ( t )( t ) b) ( k )(k ) c) ( r )(r ) d) ( m )( m ) e) ( )( ) f) ( d )(d ) 0a) ( a )( a ) b) ( )( ) c) ( m 7)( m ) d) ( k )( k ) e) ( )( ) f) ( a )(a ) a) ( )( ) b) ( m )( m ) c) ( s ) d) ( ) e) ( )( ) f) ( a )(a ) a) ( )( ) b) ( k )( k ) c) ( g )(g ) d) ( 9p )( p ) e) ( c )(c ) f) ( )( ) a) ( ) b) ( h )( h ) c) ( q )( q ) d) ( u )(u ) e) ( 0m )( m ) f) ( c )(c ) a) ( h )(h ) b) ( r )(r ) c) ( w)( w) d) ( 7t )(t ) e) ( )( 7) f) ( a ) a) 0( )( ) b) (a )( a ) c) (a )(a ) d) (r )(r ) e) ( ) f) (a )( a ) a) ( )( ) b) a(a )( a ) c) ( )( ) d) m(m )( m ) e) a (a 7)( a ) f) 7a (b )(b ) 7a) ( )(8 ) b) ( 8s )(s ) c) ( a 7)( a ) d) ( )( ) e) ( a )(a ) f) ( 7 )( ) 8a) ( 7 )( ) b) ( 9 )(8 ) c) ( )( 8) d) ( 8 )( ) e) ( c d)(c d) f) ( z)(8 z).

18 Mathematics Page of Appendi: Factoring Non-Simple Trinomials A few common techniques to factor Non-simple trinomials a Eample: Factor b c Method : B Decompositions Factor numbers: Product = & Sum = + (The are + 9 and ) = Decompose the middle term Factoring b groupings Common factoring Method : B Decomposition Formula Factor a a a numbers: Product = & Sum = + (The are + 9 and ) 9 Place those numbers inside the brackets Common factoring each bracket Method : B Trials and Errors (X-Bo) Factor = (first term) ()() + ()(-) = (middle term) (-)()= - (last term)

19 Mathematics Page of Appendi: Factoring Non-Simple Trinomials Method : B Bo Method Factor numbers: Product = & Sum = + (The are + 9 and ) Place the numbers at each corner Factor each row and each column Method : B Method Factor numbers: Product = & Sum = + (The are + 9 and ) a : : 9 9 a

20 Grade Mathematics Page of Factoring Mied Review Factor Full:. 8m m b. m + 8m +. 8k. a a + 7. a + ab + b z 8. m mn + 7n. a b c a b c.. n 8n 90n. s s k + 7k c + c d 00cd 9. p q 8p q 0 pq 0. 8m n n. 7m m s + 0s t + t 0 89a b c m 8m n + 7n 8. 8a b a b + a b 8a b 9. a + a + b + b 0. b + c + ab + ac ( ) +. ( + ) ( a + ) a a 8. ( ) + 9. ( ab ) ab + 0. ( a) + ( a)

21 Grade Mathematics Page of Factoring Mied Review Answers:. m (m ). ( + ). ( b )( b + ). ( m + ). (9k + )(k + )(k ). ( a )( a ) 7. ( a + b)( a + b) 8. ( + 7)( 7) 9. ( )( + ) 0. ( + z )( z ). ( m n)( m n). a b c(b c). n ( n 9)( n + ). (s )( s + )( s ). ( )( + )( )( + ). ( k + )( k ) 7. ( )( + ) 8. c( c + d)( c d) 9. pq ( p q)( p + q) 0. 8 ( m )( m + ). n ( n)( + n). 7( + )( + )( + )( + )( ). m (m + )(m ). ( s + 8t)( s + t). 0 ( + ). (7ab 8c )(7ab + 8c ) 8 7. ( m n )( m + n )( m n ) 8. a b (b b + a ab ) 9. ( + )( a + b) 0. ( b + c)( + a). ( + )( + ). ( )( + ). ( + )( ). ( + )( ). ( ). ( + )( + ) 7. ( a )( a + ) 8. ( )( ) 9. ( ab )( ab ) 0. ( a) ( + a)

22 Grade Mathematics Page of Functions & Relations, Domain and Range b Graph Relations and Functions Relations: Relation is a set of ordered pairs. (, ) Functions: A function is a special tpe of relation which is a set of ordered pairs in which for ever value of, there is onl one value of. Independent variable: horizontal ais variables, normall Dependent variable: vertical ais variables, normall Vertical line test If an vertical line passes through more than one point on the graph, then the relation is not a function. Function Relation Domain and Range Domain: All possible values of (independent variables); the set of the first elements in a relation. Range: All possible values of (dependent variables) within the domain; the set of second elements in a relation. Eample : Domain and Range with Ordered Pairs Given the relations a) (,) (,) (,) (7,8) b) (,) (,) (,) (,7) i) Graph the relations on the given grid. ii) Determine the domain and range for the relations; iii) Create a mapping diagram for the ordered pairs; iv) Are the representing a function or a relation? a) (,) (,) (,) (7,8) b) (,) (,) (,) (,7) Domain: Range: Domain: Range: Eample : Domain and Range with Diagrams Determine the domain and range for the following diagrams, are the represent a function or a relation? a) b) c) d)

23 Grade Mathematics Page of Functions & Relations, Domain and Range b Graph e) f) g) h) Function Notations In algebra, smbols such as & are used to represent numbers. To represent functions, we often use smbols such as f () and g (). For eample, f ( ) = read as Function of or F of Eample : Evaluating a Function Given f ( ) =, find i) f () ii) f ( ) iii) f (0.) Eample : Evaluate a Function b graph From the graph of = f () as shown. Determine i) f () ii) f (0) iii) f ( ) - Eample : Composition of Functions Given f ( ) = 7, g ( ) = +, find a) f (c) b) f ( ) c) g ( ) d) f ( ) e) f ( ) + g() f) f ( g( )) g) g ( f ( )) h) f (g(0))

24 Grade Mathematics Page of Functions & Relations, Domain and Range b Graph. State the Domain and Range of each of the given relations in the space provided. Assume that graphs drawn to the edge of the grid continue on infinitel D: D: D: D: R: R: R: R: D: D: D: D: R: R: R: R: D: D: D: D: R: R: R: R: D: D: D: D: R: R: R: R:

25 Grade Mathematics Page of Functions & Relations, Domain and Range b Graph. Given f ( ) =, g( ) =, and h ( ) =, determine each of the following in its simplest form: a ) f ( ) b ) g( ) c ) h() d ) f ( t ) e) g( t) f ) h t g ) f ( ) + h ) f ( + ) i ) f ( ) j) f ( ) k ) g() l ) h( ) + Answers ) D: <, R D: R D: 7, R D: 0, R R: < D: <, R R: = R: R, R: 8, R D: = 7 D: 7, R D: R, R R: >, R R:, R R:, R R: R D: 0, R D: R D: 0, R D:, R R: R R:, R R: 0, R R:, R D: R D: 7, R D: R D: R R:, R R:, R R: 0, R R: R a) 0 b) c) - d) t t e) t f) t t g) + h) i) j) k) undefined or i l)

26 Grade Mathematics Page of Radicals Sstem Radical epression: An epression involving the square root of an unknown ab = a b, a 0, b 0 (Entire Radical) Recall: a a m =, a 0, b > 0 ab = a b b (Mied Radical) m m ( ) b Eample Simplif from the entire radicals to mied radicals a) 7 b) 8 c) 9 Eample Simplif from the mied radicals to entire radicals a) b) c) 7 Eample Simplif a) 9 7 b) c) To simplif radical epressions, epress radicals in simplest radical form and add or subtract like radicals Eample : Adding and Subtracting Radicals Simplif a) b) + 0 +

27 Grade Mathematics Page of Radicals Sstem To multipl binomial radical epressions, use the distributive propert and add or subtract like radicals. Eample : Radical Multiplications Epand and simplif a) ( + 0) b) ( + )( ) a a = a c) ( 7 + )( 7 ) Recall: a b ( )( a + b) = a b ( a b )( a + b ) = a b Conjugate binomials: Binomials of the form a b + c d numbers. The product of conjugates is a rational number. and a b c d where a, b, c, and d are rational Eample : a) Rationalize the denominator b) 7 c) + d)

28 Grade Mathematics Page of Radicals Sstem Simplif Answers:

29 Mathematics Page of Comple Numbers Sstem Comple Numbers Comple Number: A number of the form real numbers and I is the square root of -. a + ib, where a and b are i = Imaginar Number i = Eample : Simplifing Comple Numbers Simplif a) b) c) Eample : Simplifing Comple Numbers Simplif and Evaluate a) i i b) i ( i) c) ( i ) d) 7 i e) i f) g) 0 + Eample : Adding and Subtracting Comple Numbers Simplif a) ( i ) + ( + i) b) ( i) ( i) c) ( i )( + i) Eample : Determine the quadratic equation Given the following roots. Determine the conjugate of the following and find the quadratic equation. + i a) i b)

30 Mathematics Page of Comple Numbers Sstem Eercise. Simplif. a) 7 b) c) d). Simplif. a) b) c) 0 d) Simplif. a) i b) i i. Simplif. + c) ( ) i d) ( i )( i ) a) 0 + i b) i c) 0 + d) 7 0. Simplif. a) 8 + b) c) Epand and simplif. a) ( 0 ) b) ( + )( ) c) ( ) 7. Simplif. a) 8. Simplif. b) c) + a) ( + 7i) + ( + i) b) ( + i) + i+ ( + i) c) ( 7i) ( 7 i) d) ( + i) ( + i) ( + i) e) ( 7i)( 7i) 9. Simplif. + f) ( + i)( i) d) a) i b) i c) i i d) i i 0. Simplif. a) i i b) + i c) + i + i d) ( i) ( + i) Answers a) b) c) 7 d) 7 a) i b) i c) i d) i a) - b) 0 c) 7i d) a) + i b) i c) i + 8 d) 0 a) 7 b) + 0 c) 7 + a) b) c) 8 7a) 9a) i b) b) i c) I + c) d) d) + 8a) +0i b) +i c) --i d) --i e) 8 f) i 0a) ( + ) i b) i ( i) c) d) i

31 Mathematics Page of Solving Quadratic Equations (B factoring) The general quadratic equation is of the form a b c 0. In order to solve a quadratic equation ou must epress the given equation in that form. The roots of a quadratic equation relate to the - intercepts of the corresponding quadratic function. Methods There are methods used to solve a quadratic equation. B factoring B completing the square B Quadratic formula You are going to learn how to solve a quadratic equation b factoring in this section although ou must be familiar with all of them in a long run. Eample : Solving Equations Given Factored form a) 0 0 b) 0 c) 0 d) 0 Eample : Solving Equations b factoring a) 0 0 b) 0 c) 0 d) 0 e) f) g) 0 h) 0 7

32 Mathematics Page of Solving Quadratic Equations (B factoring) ) Mied Factoring - Factor Full: a) 8 b) d) g) j) l) n) p) s z e) 9 00 h) p q 8p q 0pq k) 0s t t m) a b c o) 8a b a b a b 8a b ) Solve b factoring: 7 a ab b c) m mn 7n f) 0 i) m 8 8m n 7n 9 a b c a b c c c d 00cd a) 8m m = 0 b) b = 0 c) m 8m = 0 d) 8k = 0 e) a a = 0 f) 8 = 0 g) n 8n 90n = 0 h) s s = 0 i) k 7k 70 = 0 j) 8m 7 = 0 k) Answers: 0n n = 0 l) 7m m = 0 a) ( ) b) ( a b)( a b) c) ( 7)( 7) 9 9 d) ( z )( z ) e) ( m n)( mn) f) a b c(b c) g) ( )( )( )( ) h) ( )( ) i) c( c d)( c d) j) pq ( p q)( p q) k) 7( )( )( )( )( ) l) ( s 8t)( s t) m) 0 ( ) 0 0 n) (7ab 8c )(7ab 8c ) o) ( m n )( m n )( m n ) p) a b (b b a ab ) a) m (m ) 0 m 0; m e) ( a )( a ) 0 a ; a h) (s )( s )( s s ; s n 0; n b) ( b )( b ) 0 b ; b f) ( )( ) 0 ) 0 k) n( n)( n) 0 0; 0; ; i) ( k k l) m )( k c) ( m ) & k (m m 0; m m ) 0 )(m 0 ) 0 d) (9k )(k )(k ) 0 k g) n ( n 9)( n ) 0 n 0; n 9; n j) 8( m )( m ) 0 m

33 Mathematics Page of Review of Quadratic Functions (All tpes) Functions: A function is a special kind of relation in which for ever value of there eists onl one value of. Details of functions will be discussed in the later sections. Recall: = a + b + c is a standard form of a quadratic function, also called a parabola. The three forms of quadratic functions are as follow: Standard Form Factored Form Verte Form = a + b + c = a( + m)( n) = a( p) + q intercept: = c intercepts: = m & = n Verte: ( p, q) To find -intercepts; let = 0 Rule of transformation: b ± b ac For a quadratic in: - B quadratic formula = a = a( p) + q - B factoring - B isolating after the verte form is done. (, ) ( + p, a + q) - intercepts Also called Zeros or Roots, can be determined b letting the functions = 0. -m n Ais of smmetr: = average of the roots Or the value of the verte. c (p, q) Directions of Opening If a > 0, opens up (Concaves up); If a < 0, opens down (Concaves down). = p Domain: R Range: if a > 0, if a < 0, q q Eample : Reviewing Quadratic in Standard form Given the quadratic function: f ( ) = + a) Determine the intercepts for b factoring: b) Determine the ais of smmetr and the verte.

34 Mathematics Page of Review of Quadratic Functions (All tpes) Eample : Transformation of Quadratic Function a) Convert = + into verte form b completing the square. b) Determine the roots b isolating from the verte form. c) Determine the rules of transformations and describe the transformations. d) Sketch the parabola and determine the domain and range. Eample : Transformation of Quadratic Function a) Convert = + into verte form b completing the square. b) Determine the roots b Quadratic formula. c) Determine the rules of transformations and describe the transformations. d) Sketch the parabola and determine the domain and range.

35 Mathematics Page of Review of Quadratic Functions (All tpes) Eample : Converting to Verte form b Completing the square a) Convert = into verte form b completing the square. b) Determine the roots b isolating from the verte form. c) Determine the rules of transformations and describe the transformations. d) Sketch the parabola and determine the domain and range. Recall: Nature of roots (Discriminant) = a + b + c If b ac > 0, different real roots If b ac = 0, equal real roots If b ac < 0, No real roots Or imaginar roots

36 Mathematics Page of Review of Quadratic Functions (All tpes) Eercise. Which of the following trinomials are perfect square trinomials? (circle the letter) (tr to discover a pattern from the answers above) a ) b ) + 9 c ) + 9 d ) 9 e ) f ) + + g ) + + h ) + i ) ) 9 + m ) n ) j k ) l ) + 9. Fill in the blank with the appropriate number to make each a perfect square trinomial. a ) + b) 0 c) 0 d) + e ) f ) + g ) + h ) + +. Beside each of the trinomials in #, write its factored form. Epress our answer as ( + p) or ( p).. Convert each of the following into verte form b completing the square. a) = + b) = 0 c) = + 8 d) = + 8 e) = f) = g) = + + h) = i) = + j) = + k) = +. Without graphing each function, convert the function into verte form, then state whether it has a maimum or a minimum value. State the maimum or minimum value of the function and the value of when it occurs a) = + + b) = + 8 c) = + d) = e) i) = f) = j) = + = + + g) = + 8 h) =. Solve each of the following equations using an appropriate method: a. + + = 0 b. = c. + = d. = e. + = 0 f = 0 g. + ( + ) = ( + ) h. 8 = 0 i. ( )( + )( + ) =

37 Mathematics Page of Review of Quadratic Functions (All tpes) 7. For each of the following quadratic functions i) Factor the trinomial to determine the zeros, ii) Determine the equation of the ais(line) of smmetr; iii) Determine the verte of the parabola; iv) Sketch the graph. a) = + 8 b) = + c) = + 8. Solve each of the following b using quadratic formula. Leave our final answer in square root form. Write the appropriate steps. a) = 0 b) = 0 c) + = 0 9. Fill in the blanks in the following chart. equation = ( ) + = = ( ) + = verte direction of opening equation of ais of smmetr verte a maimum or a minimum point shape compared to = state the ma or min value -intercepts -intercept 0. Determine the optimum (Ma or Min) value and what is the value when it occurs b factoring a) = b) = 0 0 c) = + d) = + 0 e) = f) = + 0

38 Mathematics Page of Review of Quadratic Functions (All tpes) Answers ) a c g h k a) b) 00 c) d) e) f) g) h) a) ( + ) b) ( 0) c) ( ) d) ( + ) e) ( + ) f) ( ) g) ( ) h) ( +) a) = ( + ) 9 b) = ( ) c) = ( + 9) 8 d) = ( + ) 9 e) = ( ) f) = ( + ) g) i) = ( + ) + 9 j) = ( + ) 7 k) 7 = + h) = = + a) = ( + ) 7 Min value = -7 at = - b) ( ) + c) = ( ) Min value = at = d) = ( + ) 8 = + Min value = - at = - + Ma value = 8 at = - e) = ( + ) + Ma value = at = - f) = + + Min value = at = 8 8 g) = ( ) Ma value = 0 at = h) = ( ) i) = ( ) Ma value = - at = j) ( ) Min value = - at = = Min value = 0 at = ± ± 7 a) -, - b) c) -7, d) 0, e) No Soln f) No soln g), - h) i) 7a) i), ii) = iii) (, -) b) i), - ii) = - iii) (-,-) c) i), ii) =. iii) (.,.) 8a) 7 ± b) ± c) ± 9 a) (,); down; = ; Ma; Stretch verticall b factor of, reflects about ais; ; ± ; -7 a) (0, 0); down; = 0; Ma; Stretch verticall b factor of 0., reflects about ais; 0; 0; 0 b) (, 0); up; = ; Min; Same; 0; ; 9 c) (-, -); up; = ; Min; Same; -; - or 0; 0 0a) Min value is 9 when = c) Ma value is when = 8 7 b) Min value is when = d) Ma value is 0 when = e) Min value is 8 when = 7 0 f) Ma value is when =

39 Mathematics Page of Graphing of Quadratic Functions (All tpes) Functions: A function is a special kind of relation in which for ever value of there eists onl one value of. Details of functions will be discussed in the later sections. Recall: a b c is a standard form of a quadratic function, also called a parabola. The three forms of quadratic functions are as follow: Standard Form Factored Form Verte Form a b c a m n a p q intercept: c intercepts: m & n Verte: p, q To find -intercepts; let = 0 Rule of transformation: b b ac For a quadratic in: - B quadratic formula a a p q - B factoring - B isolating after the verte form is done., p, a q - intercepts Also called Zeros or Roots, can be determined b letting the functions = 0. -m n Ais of smmetr: = average of the roots Or the value of the verte. c (p, q) Directions of Opening If a > 0, opens up (Concaves up); If a < 0, opens down (Concaves down) = p Domain: R Range: if a > 0, q Families of Quadratic Functions if a < 0, q Equations that define quadratic functions can look different, but their graphs can have similarities. A group of parabolas that share a common characteristic (similarit) is called a famil of parabolas. Eample : Families of Quadratic Functions: Determine the similarities of the following group of quadratic functions. Functions Forms Similarities a) f g b) m 0 n c) r s 9

40 Mathematics Page of Graphing of Quadratic Functions (All tpes) Eample : Determine the Quadratic Function Determine the equation of the quadratic function in verte form that passes through (-, 0) if its zeros are & -. Use the information to show a diagram and determine the domain and range. Eample : Determine the Quadratic Function Determine the equation of the quadratic function in standard form with verte (-, -) and passing through (-,-). Use the information to show a diagram and determine the domain and range. Eample : Determine the Quadratic functions given diagram Determine the equation of the quadratic function in standard form with the given parabolas. Determine the domain and range. a) b)

41 Mathematics Page of Graphing of Quadratic Functions (All tpes) Eercise: ) Match the relation with the proper diagram ) Graph each of the following using the ke features. (Don t forget to label!!) a) b) c) d) 7 e) 7 f) ) ) ) ) d c b a ) ) ) ) h g f e ) ) ) ) l k j i

42 Mathematics Page of Graphing of Quadratic Functions (All tpes) a c d - f. Fill in the blanks for each of the following quadratic functions. a) ( ) verte direction of opening b) ( ) ais of smmetr -intercept c) ( ) min/ma value shape compared to (circle appropriate tpe) graph of d) verte ais of smmetr e) ( ) direction min/ma point of opening (circle appropriate tpe) f) ( ) -intercept verte g) direction of opening min/ma point (circle appropriate tpe) h) -intercepts (if an) verte

43 Mathematics Page of Graphing of Quadratic Functions (All tpes). Draw accurate graphs of the following functions. Clearl indicate the ke points used. a ) ( ) b). Determine the equations of the following parabolas for the given information. a) verte ( 0,0) passing through the point (,) b) verte, ) and -intercept ( c) ais of smmetr -, with a maimum value of, with the same shape as d) minimum point of (, 9) with -intercepts of, Answers ) i k a f d c j l b e h ) a e b f c d a) b) c) d) 9

44 Mathematics Page of Quadratic (Ma & Min) Problems Optimization Problems related to Quadratics: Optimum value is the - value of the verte; which is the result of the problem. Two methods to find the verte algebraicall: Method : Convert the quadratic relations into verte form = a( p) + q If a > 0, minimum value is q occurs when = p. If a < 0, maimum value is q occurs when = p. A B Method : Determine the ais of smmetr from the zeros of the quadratic relations and use it to determine the optimum value of the quadratic. Optimum Point & Optimum Value Maimum Points: A & B Maimum Value: Eample A glassworks compan makes lead-crstal bowls, that creates a dail production cost C given b C = 0.b 0b + 0 where b is the number of bowls made. a) How man bowls should be made to minimize the cost? b) What is the cost if this man bowls are made? Eample Two numbers have a difference of. Find the numbers if the result of adding their sum and their product is a minimum.

45 Mathematics Page of Quadratic (Ma & Min) Problems Eample A rectangular fence is to be built around a plaground; one side of the plaground is against the school. If there is a00m of fencing available, what dimensions would create the largest plaground area? School Eample A theatre seats 000 people and charges $0 for a ticket. At this price, all the tickets can be sold. A surve indicates that if the ticket price is increased, the number sold will decrease b 00 for ever dollar of increase. What ticket price would result in the greatest revenue?

46 Mathematics Page of Quadratic (Ma & Min) Problems Eercise. The general equation of a thrown object is given b h = h0 + v0t t, where the value of h 0 represents the initial height and the value of v 0 represents the initial speed of the object. a) Determine the equation representing the height of a rock that is thrown upward from a cliff that is m high, at an initial speed of 0 m/s. b) Determine the maimum height of the rock.. A ball is thrown from an apartment building. Its height, in metres, after t seconds, is given b h = t + 0t +. a) State the initial height of the ball. b) Determine the maimum height of the ball. c) Determine the length of time it takes for the ball to reach that height.. Two numbers differ b 8. Their product is to have the least value possible. Determine the numbers.. The sum of the base and the height of a triangle is cm. What is the greatest possible area for a triangle having this propert.. A rectangular lot is bordered on one side b a stream and on the other three sides b fencing. If there is 00 metres of fence available, determine the dimensions of the lot with the greatest area.. A rectangular field is enclosed b a fence and divided into two lots b another section of fence parallel to two of its sides. If the 00 metres of fence that is used must enclose a maimum area, what are the dimensions of the field? 7. A fence is to be built around the area shown in the diagram. Determine the values of and that would produce a minimum area if the perimeter is 00 metres. Tetbook: P. # -8, 0-,-8 Answers ) a) h = t + 0t + b) 0 a) b) 0 c) ) & - ) 8. ) 0 00 ) ) = = 9 9

47 Mathematics Page of Quadratic (Ma & Min) Problems (Da ) Optimization Problems related to Quadratics: Optimum value is the - value of the verte; which is the result of the problem. Two methods to find the verte algebraicall: Method : Convert the quadratic relations into verte form = a( p) + q If a > 0, minimum value is q occurs when = p. If a < 0, maimum value is q occurs when = p. Method : Determine the ais of smmetr from the zeros of the quadratic relations and use it to determine the optimum value of the quadratic. b b ac Method : B formula!! Verte :, a a This can be proved b completing the squares of = a + b + c!!! Eample Find the minimum product of two numbers whose difference is 8. Eample A strip of sheet metal 0 cm wide is to be made into a tunnel b turning strips up verticall along two sides. How man centimetres should be turned up at each side to obtain the greatest carring capacit? 0 cm

48 Mathematics Page of Quadratic (Ma & Min) Problems (Da ) Eample A magazine producer can sell 00 of her magazines at $.00 each. A marketing surve shows her that for ever $0.0 she increases the price, she will lose 0 sales. What price should she set to obtain the greatest revenue? Eample The student council plans to run the annual talent show to raise mone for charit. Last ear, ticket sold for $ each, and 00 people attended. The student council has determined that, for ever $ increase in price, the talent show attendance would decrease b 0 people. If there is a cost for $8 per person, what ticket price would maimize the profit and what is the maimum profit? Profit = Revenue - Cost

49 Mathematics Page of Quadratic (Ma & Min) Problems (Da ) Eercise. i. Write the equation in verte form. ii. Write the equation of the ais of smmetr. iii) Write the coordinates of the verte. iv) What is the maimum or minimum value of the function? v) Sketch the graph of the functions? a) f( ) = + b) gh ( ) = h h 8 c) f( b) = 0.7(. b)(. + b ) d) p ( ) = ( )( + ). The sum of two numbers is 0. What is the largest product of these numbers?. A football is punted into the air. It s height h, in metres, after t seconds is given b h( t) = +.t.9t. a) Find the maimum height of the ball b completing the square. Round our answer to the nearest hundredth of a metre. b) When does the ball reach its maimum height?.a) An orchard owner has maintained records that show that, if apple trees are planted in one acre, then each tree ields an average of 00 apples. The ield decreases b 0 apples per tree for each additional tree that is planted. How man trees should be planted for maimum total ield? b) The cost of maintaining each tree is $.0 and the owner can epect to sell his apples for cents each. How man trees should he plant for maimum profit?. A -kg steer gains. kg/da and costs 80 cents/da to keep. The market price for beef cattle is $./kg, but the price falls b cent/da. When should the steer be sold to maimize profit?. Mark is designing a pentagonal-shaped pla area for a dacare facilit. He has 0 m of nlon mesh to enclose the pla area. The triangle in the diagram is equilateral. Find the dimensions of the rectangle plus the triangle, to the nearest tenth of a metre, that will maimize the area he can enclose for the pla area. 7. The diagram of a practice field shows a rectangle with a semicircle at each end. The track-and-field coach wants two laps around the field to be 000 m. The phsical education department needs a rectangular field that is as large as possible. a) Determine the dimensions of the track that will maimize the entire enclosed area. Do these dimensions meet the needs of the track coach and the phsical education department? Eplain. b) If onl the rectangular portion of the field is maimized, can the track team run the 00-m dash along a straight part of the track? Justif our answer. Answers a) i) ( ) f = + ii) = iii), iv) Min: g h = h ii) h = iii) 7, iv) Min: f b = 0.7 b +.0. ii) =. 0 b) i) ( ) c) i) ( ) ( ) b iii) (.0,.7) iv) Ma:. 7 d) i) p ( ) = + ii) = iii), 8 iv) Ma: 8 8 ) a).m b). sec a) 7 b) ) das ) Triangle: 7.0m Rectangle:.m 7.0m 7a) Radius of semi-circle: 79. m

50 Mathematics Page of Quadratic Word Problems (Roots) Another word problem related to Quadratics is when the roots of the quadratics are the answer: Recall: The three forms of quadratic functions are as follow: Standard Form Factored Form Verte Form a b c a m n a p To find -intercepts (roots) Let = 0 - B quadratic formula b b ac a - B factoring - B isolating after the verte form is done. q Eample : Area Problem A rectangle is cm longer than it is wide. If the area is 9 cm, determine its dimensions. Eample : Number Problem The sum of the squares of two consecutive numbers is. Determine the numbers.

51 Mathematics Page of Quadratic Word Problems (Roots) Eample : Border Problem A rectangular lawn measuring 8 m b m is surrounded b a rectangular flower bed of uniform width. The combined area of the lawn and the flower bed is m. What is the width of the flower bed?

52 Mathematics Page of Quadratic Word Problems (Roots) Eercise. Two squares whose side lengths are consecutive odd integers have a total area of 0 cm. Determine their total perimeter.. A right triangle has a perimeter of metres. Its hpotenuse has a length of 0 centimetres. Determine the length of the other two sides.. A rectangular bo is 0 cm high and is twice as long as it is wide. If the surface area is 00 cm, determine the volume of the bo.. A rectangular nuclear-waste facilit is 00 metres long and 70 metres wide. A safet zone of a uniform width of more than 0 metres must be constructed around the facilit. Determine the width of the strip, to the nearest metre, if the total area of the facilit and the strip is square metres.. Two bos agree to row a rectangular lawn that is 80 metres b 0 metres. One bo starts on the outside of the lawn cutting a uniform strip around the lawn, while the other bo begins at the centre and cuts outward. If each bo is to cut an equal share of the lawn, how wide a strip will the first bo need to cut?. The length of a rectangle is metres more than the width. If the area is m, what are the dimensions of the rectangle? 7. Two numbers differ b. If the numbers are squared and then added, the result is. What are the numbers? 8. The hpotenuse of a right triangle is 7 cm long. The net longest side is 7 cm longer than the third side. Determine the unknown lengths. 9. An area rug has a central patterned section that is m b m. There is a plain border of uniform width surrounding this patterned section. The total area of the rug is m. Determine the width of the border. 0. A picture that measures 0 cm b cm is to be surrounded b a matte before being framed. The width of the matte is to be the same on all sides of the picture. The area of the matte is to be twice the area of the picture. What is the width of the matte?. Helen owns a campground that has a rectangular swimming pool measuring 0 m b 0 m. She wants to put a wooden deck of uniform width around the pool. She knows that the cost of the deck will be $0 per square metre and has $90 in her budget to pa for the deck. What is the widest that the deck can be?. Show that it is impossible to bend a 0 cm length of wire into a rectangle with an area of 0 cm. Answers ) cm ) 0cm & 0cm ) 000cm ) 07.7m ) 0m ) m b 9m 7) - & - or & 8) 8 cm & cm 9) 0.m 0).cm ) m

53 Mathematics Page of Sum and Product of roots to Quadratics There are simple relations between the roots of a quadratic equation and its coefficients. Let the equation a b c, a 0,, have the roots Eample Find the sum and product of the roots of the equation a) 7 0 b) 0

54 Mathematics Page of Sum and Product of roots to Quadratics Eample One root of 7 0 is. Find the other roots. Eample Check that and 7 are the roots of 0

55 Grade Mathematics Quadratic Problems (Etra). An enclosure is constructed with the shape shown and with a perimeter of 00 m. What are the values of and so that the area of the enclosure is a maimum?. Find two numbers that has difference of 9 and whose product is a minimum.. What are the values of two positive numbers whose sum is 0 and whose product ields a maimum? Name. In a newspaper contest a problem is posed. The last two numbers for a combination to open a safe add up to. As an additional clue, the product of the two numbers is a maimum. What are the last two numbers?. What is the maimum area of a rectangular sand lot that can be enclosed b 00 m of fencing?. A parking lot is to be fenced in on three sides with the fourth side bounded b a building. If 00 m of fencing are available, what is the maimum area of the parking lot? 7. An isosceles triangle has base m and height m. Find the maimum area of the rectangle placed as shown in the diagram. 8. From preliminar computer results, it is found that two required numbers have a sum of. Before the computer breaks down, it indicates that the product of the numbers is a maimum. What are the numbers? m m 9. The Environment Group wants to designate an area as a natural habitat. One side of the rectangularshaped area is a large lake. Not including this side the group is allowed to have the remaining sides total 8 km. What dimensions will ensure a maimum area for the natural habitat? 0. Studies have shown that 00 people attend a high school basketball game when the admission price is $.00. In the championship game admission prices will increase. For ever 0 cents increase 0 fewer people will attend. What price will maimize receipts?. The Transit Commission s single-fare price is 0 cents cash. On a tpical da approimatel persons take the transit and pa the single-fare price. To reflect higher costs, single fare prices will be increased, but surves have shown that ever cents increase in fare will reduce ridership b 000 riders dail. What single-fare price will maimize income for the commission based on single fares?. Slacks Incorporated sold 000 pairs of slacks last month at an average price of $ each. The store is going to increase prices in order to increase profits. Sales forecasts indicate that sales will drop b 00 for ever dollar increase in price. On the average, the compan pas $ for each pair of slacks it sells. What price will maimize profits?. An auto parts store currentl sells 00 spark plug packages each week at a price of $.0 each. To increase sales and reach more customers the part outlet decides to reduce the price of the package, knowing that ever 0 cents decrease in price will result in more sales. What price will maimize total revenue?. Tri Electronics sells radios for $0 each. 0 radios are sold dail. A surve indicates that a price raise of $.00 will cause the loss of one customer. If the cost of producing the radios is $8.00, how much should the compan charge to maimize the profit?. A compan selling MP Plaers for $80 each sells 0 each da. A surve indicates that for each dollar the price is raised one customer will be lost. The cost of making the MP plaers is $ each. How much should the compan charge to maimize the profit?

56 Grade Mathematics Quadratic Problems (Etra) Name. Two numbers differ b and the sum of their squares is 08. What are the numbers? 7. A package designer wants to protect an epensive square book with a cm wide cardboard rim. The area of the rim of the front cover is equal to the area of the front cover of the book. What is the length of the book? 8. The perimeter of a right triangle is 0 cm. The lengths of sides of a right triangle, in cm, are shown. Find the lengths of all sides of the triangle. 9. In travelling from Arcola to Beamsville ou can use either the super highwa or connect through Pasqua as shown. On the super highwa ou can travel 0 km/h faster, and take h less time. What is the average speed for each route? 0. When two consecutive integers are squared and the squares added their sum is. What are the possible numbers?. A right-angled triangle has a height 8 cm more than twice the length of the base. If the area of the triangle is 9 cm, find the dimensions of the triangle. 0 km Beamsville Pasqua. For a certain model of a jet plane, aerodnamics requires that the tail have the profile of a right-angled triangle with area 7. m. If the height must be m less than the base length for attachment purposes, calculate the base length of the tail.. A rectangular solar heat collecting panel has a length m more than its width. If the area of the solar panel is 8 m, how long is the panel? w Super Highwa 70 km 0 km w Arcola. A right-angled triangle has a perimeter of 0 cm. If the hpotenuse is 0 cm, find the lengths of the other two sides.. A matte of uniform width is to be placed around a painting so that the area of the matted surface is twice the area of the picture. If the outside dimensions of the matte are 0 cm and 0 cm, find the width of the matte.. A square swimming pool with a side measuring m is to be surrounded b a uniform rubberized floor covering. If the area of the floor covering equals the area of the pool, find the width of the rubberized covering. 7. In the annual 0 km charit Walk-A-Thon, Mark and Tina leave at the same time. Tina walks 0.8 km/h faster than Mark, but stops to have her feet taped, thus losing 0. h. Even with this dela Tina finishes the race h before Mark. a) How fast was each person walking? b) How long did it take each to walk the course? Answers: ) =., =. )., -. ) 0, 0 ), ) 00 m ) m 7) m 8) ) km b km 0) $.0 ) $.0 ) $9.0 ) $.0 ) $.00 ) $97.00 ) 8 & ; -8 & - 7). cm 8) 0 cm, cm, cm 9) Ordinar 70 km/h Super 90 km/h 0) & ; - & - ) base 8 cm, height cm ) m ) 7 m ) 0 cm, 0 cm ) 0 cm ). m 7a) Mark km/h, Tina.8 km b) Mark h, Tina h

57 Mathematics Page of Intersection of Linear and Non-linear Relations A linear-quadratic sstem can be solved algebraicall b a method we used for solving linear sstems. The linear equation is solved for either variable, and the epression obtained substituted in the quadratic equation. A linear relation and a non-linear relation can intersect at zero, one, or two points. A linear-quadratic sstem of equations can have no solution, one solution, or two solutions. Eample : Finding Points of Intersection of a Line and a Circle Find the coordinates of the points of intersection of the line and the circle. a) graphicall b) algebraicall Eample : Finding Points of Intersection of a Line and a Parabola Find the coordinates of the points of intersection of the parabola ( ) a) the line b) the line and Eample : Finding Points of Intersection of a Line and a Reciprocal Function Find the coordinates of the points of intersection of the reciprocal function and the line 0

58 Mathematics Page of Intersection of Linear and Non-linear Relations Eample : Problem Solving A sk diver jumped from an airplane and fell freel for several seconds before releasing her parachute. Her height, h, in metres, above the ground at an time is given b: h.9t 000 before she released her parachute, and h t 000 after she released the parachute. a) How long after jumping did she release her parachute? b) How high was she above the ground at that time? c) If the parachute released, when will the sk diver landed on the ground? Eercise. Solve each sstem of equations. a) b) 7 0 c) d). Solve each sstem of equations. Round answers to the nearest tenth, if necessar. Include a sketch with our solution. 0 a) 0 b) 0 c) 8 d). Solve each sstem of equations. Round answers to the nearest tenth, if necessar. Include a sketch with our solution. 7 a) b) c) One point moves in such a wa that its distance from the -ais is three times its distance from the - ais, and another moves along the circle whose radius is and whose centre is the origin. Where do the two paths cross? Answers: a) (,) b) (,) &(,) c) (,) d) No P.I. a) (-.,. & -.,0) b) (, -) c) (, -) d) (0, -) & (, -9) a) (.,.) & (-, ) b) (0,-) & (-,) c) No P.I. ) (.7,.8) & (-.7, -.8) Homework P. 7 #,,7,,9,0