Lesson 4.6 Exercises, pages
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1 Lesson 4.6 Eercises, pages A 3. Identif the -intercepts of the graph of each quadratic function. a = ( + 3( - b = ( - 4( - 5 The -intercepts are: 3 and The -intercepts are: 4 and 5 c = ( + 1( + 7 d = -( + 5 The -intercepts are: 7 and 1 The -intercepts are: 5 and 0 e = -( + 5(3-3 f = (4-1(5 + 1 The -intercepts are: 5 and 1 The -intercepts are: 1 and Determine the zeros of each quadratic function. a = b = ( 3( 4 The zeros are: 4 and ( 7 ( 7 ( 1( 7 The zeros are: 1 and 7 P DO NOT COPY. 4.6 Analzing Quadratic Functions of the Form = a + b + c Solutions 7
2 B 5. Determine the intercepts, the equation of the ais of smmetr, and the coordinates of the verte of the graph of each quadratic function. a = b = The -intercept is 3. Factor the equation. 0 3 ( ( 8( The -intercepts are: and 8 The mean of the intercepts is: 8 5 So, the equation of the ais of smmetr is: 5 Substitute 5 in 0 3 (5 0( The coordinates of the verte are: (5, 18 The -intercept is 6. Factor the equation ( ( 3( 6 3 The -intercepts are: 3 and The mean is: 4.5 So, the equation of the ais of smmetr is: 4.5 Substitute 4.5 in (4.5 3( The coordinates of the verte are: (4.5, Sketch a graph of each quadratic function. List the characteristics ou used. a = Discriminant is: ( 1 4( 3( Complete the square a ( b 9 The graph of opens down and is congruent to the graph of 3. On a grid, mark a point at the verte: ( 3.5, Use the step pattern. Multipl each vertical step b b = Discriminant is: ( 0 4(5( Complete the square ( ( 3 The graph of opens up and is congruent to the graph of 5. On a grid, mark a point at the verte: (, 3. Use the step pattern. Multipl each vertical step b Analzing Quadratic Functions of the Form = a + b + c Solutions DO NOT COPY. P
3 7. Determine the coordinates of the verte of the graph of = b factoring and b completing the square. Which strateg do ou prefer? Wh? Completing the square: ( ( ( ( ( The coordinates of the verte are: (0.5, Factoring: ( ( 1.5( 1.0 The -intercepts are: 1.5 and 1. The mean of the intercepts is: So, the equation of the ais of smmetr is: 0.5 Substitute 0.5 in (0.5.5( The coordinates of the verte are: (0.5, I prefer the completing the square strateg because I can find the coordinates of the verte in one step. 8. a The graph of a quadratic function passes through A(1, 8 and has -intercepts and 5. Write an equation of the graph in factored form. Use a( 1 ( Substitute: 1 and 5 a( ( 5 Substitute for A(1, 8. 8 a(1 ( a a In factored form, the equation is: ( ( 5 b The graph of a quadratic function passes through B(1, -7, and the zeros of the function are -6 and -1. Write an equation of the graph in general form. Use a( 1 ( Substitute: 1 6 and 1 a( 6( 1 Substitute for B(1, 7. 7 a(1 6( a a 0.5 In factored form, the equation is: 0.5( 6( 1 Epand to write the equation in general form. 0.5( 6( 1 0.5( P DO NOT COPY. 4.6 Analzing Quadratic Functions of the Form = a + b + c Solutions 9
4 9. For each graph of a quadratic function, write the equation in the form given. a factored form b general form g( f( 6 0 -intercepts: 5, 4 passes through (3, 4 Use a( 1 ( Substitute: 1 5 and 4 a( 5( 4 Substitute for (3, 4. 4 a(3 5( a a 0.5 In factored form, the equation is: 0.5( 5( 4 -intercepts: 3, 5 passes through ( 1, 10 Use a( 1 ( Substitute: 1 3and 5 a( 3( 5 Substitute for ( 1, a( 1 3( a a 1.5 In factored form, the equation is: 1.5( 3( 5 Epand to write in general form: 1.5( The cross section of a satellite dish is parabolic. The parabola has a maimum depth of 7.5 cm and a width of 50 cm. 50 cm 7.5 cm a Determine an equation to model the parabolic dish. Sample response: Place one end of the dish at the origin. Since the dish is 50 cm wide, the other end of the dish has coordinates (50, 0. The maimum depth of the dish is 7.5 cm, so the verte has coordinates (5, 7.5. Use the factored form: a( 1 ( Substitute: 1 0 and 50 The equation becomes: a( 50 Substitute: 5 and a(5( a a 0.01 So, an equation that models the parabolic dish is: 0.01( Analzing Quadratic Functions of the Form = a + b + c Solutions DO NOT COPY. P
5 b How deep is the dish 10 cm from its edge? Use the equation: 0.01( 50 Substitute: (10( Since the point (10, 4.8 is 4.8 units below the -ais, the dish has a depth of 4.8 cm. 11. The top of a window is a parabolic arch. The dimensions are shown on the diagram. Additional supports are to be added halfwa between each edge and the centre of the window. What is the length of each support? m 3 m 3 m Place one end of the arch at the origin. Since the window is 3 m wide, the other end of the arch has coordinates (3, 0. The maimum height of the arch is m, so the verte has coordinates (1.5,. Use the factored form: a( 1 ( Substitute: 1 0 and 3 The equation becomes: a( 3 Substitute: 1.5 and a(1.5( a a 8 9 So, an equation that models the parabolic arch is: 8 ( 3 9 The centre of the window is the ais of smmetr: 1.5 So, the vertical line halfwa between the centre of the window and the origin is: 1.5, or 0.75 Substitute (0.75( The two supports are congruent. Each support has length 1.5 m. 1. A student sas that the quadratic function + 15 = ( + ( + 4 has zeros - and -4. Eplain the student s error and determine the zeros. The equation is not written in factored form: a( 1 ( 15 ( ( 4 ( ( ( 7( 1 So, the zeros of the equation are 7 and 1. P DO NOT COPY. 4.6 Analzing Quadratic Functions of the Form = a + b + c Solutions 31
6 13. Sketch a graph of this quadratic function. = Discriminant is: (0.75 4( 0.5( Since is a perfect square, the equation factors ( ( 4( 1 The -intercepts are: 1 and 4. The mean of the intercepts is: So, the equation of the ais of smmetr is: 1.5 Substitute 1.5 in ( ( The coordinates of the verte are: (1.5, The graph of is congruent to the graph of C 14. The factored form of the equation of a quadratic function is = a( -. The verte form is = a( - p 1 ( - + q. Epress p and q in terms of 1 and. From the factored form of the equation, a( 1 (, the -intercepts of the graph of the equation are 1 and. The mean of the 1 -intercepts is, so the equation of the ais of smmetr is 1 1. So, the -coordinate of the verte is:. To determine the -coordinate of the verte, substitute 1 in the factored form of the equation. a( 1 ( aa 1 aa 1 1 ba 1 b aa 1 ba 1 b aa 1 ba 1 b a 4 ( 1 1 ba 1 b From the verte form of the equation, a( p q, the coordinates of the verte are (p, q. Therefore, p 1 and q a. 4 ( Analzing Quadratic Functions of the Form = a + b + c Solutions DO NOT COPY. P
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