CHAPTER 2 - QUADRATICS
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1 CHAPTER 2 - QUADRATICS
2 VERTEX FORM (OF A QUADRATIC FUNCTION) f(x) = a(x - p) 2 + q Parameter a determines orientation and shape of the parabola Parameter p translates the parabola horizontally Parameter q translates the parabola vertically STANDARD FORM (OF A QUADRATIC FUNCTION) f(x) = ax 2 + bx + c Parameter a determines orientation and shape of the parabola Parameter b influences the position of the graph Parameter c determines the y-intercept of the graph
3 VERTEX FORM (OF A QUADRATIC FUNCTION) f(x) = a(x - p) 2 + q STANDARD FORM (OF A QUADRATIC FUNCTION) f(x) = ax 2 + bx + c You can expand f(x) = a(x - p) 2 + q and compare the resulting coefficients with the standard form f(x) = ax 2 + bx + c, to see the relationship between the parameters of the two forms of a quadratic function.
4 VERTEX FORM (OF A QUADRATIC FUNCTION) f(x) = a(x - p) 2 + q STANDARD FORM (OF A QUADRATIC FUNCTION) f(x) = ax 2 + bx + c
5 VERTEX FORM (OF A QUADRATIC FUNCTION) f(x) = a(x - p) 2 + q STANDARD FORM (OF A QUADRATIC FUNCTION) f(x) = ax 2 + bx + c p = b 2a q = c ap 2 vertex (p,q) vertex ( b 2a,c ap2 )
6 For each graph of a quadratic function, identify the following: the direction of opening the coordinates of the vertex the maximum or minimum value the equation of the axis of symmetry p = b 2a the x-intercepts and y-intercept the domain and range opens upward vertex: (0, 0) minimum value of y of 0 when x = 0 axis of symmetry: x = 0 y-intercept occurs at (0, 0) and has a value of 0 x-intercept occurs at (0, 0) and has a value of 0 domain: all real numbers, or {x x R} range: all real numbers greater than or equal to 0, or {y y 0, y R} f(x) = ax 2 + bx + c q = c ap 2 c = y - intercept
7 For each graph of a quadratic function, identify the following: the direction of opening the coordinates of the vertex the maximum or minimum value the equation of the axis of symmetry p = b 2a the x-intercepts and y-intercept the domain and range opens upward vertex: (1, -1) minimum value of y of -1 when x = 1 axis of symmetry: x = 1 y-intercept occurs at (0, 0) and has a value of 0 x-intercepts occur at (0, 0) and (2, 0) and have values of 0 and 2 domain: all real numbers, or {x x R} range: all real numbers greater than or equal to -1, or {y y -1, y R} f(x) = ax 2 + bx + c q = c ap 2 c = y - intercept
8 For each graph of a quadratic function, identify the following: the direction of opening the coordinates of the vertex the maximum or minimum value the equation of the axis of symmetry p = b 2a the x-intercepts and y-intercept the domain and range opens downward vertex: (1, 9) maximum value of y of 9 when x = 1 axis of symmetry: x = 1 y-intercept occurs at (0, 8) and has a value of 8 x-intercepts occur at (-2, 0) and (4, 0) and have values of -2 and 4 domain: all real numbers, or {x x R} range: all real numbers less than or equal to 9, or {y y 9, y R} f(x) = ax 2 + bx + c q = c ap 2 c = y - intercept
9 For each graph of a quadratic function, identify the following: the direction of opening the coordinates of the vertex the maximum or minimum value the equation of the axis of symmetry p = b 2a the x-intercepts and y-intercept the domain and range opens upward vertex: (3, 7) minimum value of y of 7 when x = 3 axis of symmetry: x = 3 y-intercept occurs at (0, 25) and has a value of 25 no x-intercepts domain: all real numbers, or {x x R} range: all real numbers greater than or equal to 7, or {y y 7, y R} f(x) = ax 2 + bx + c q = c ap 2 c = y - intercept
10 For each graph of a quadratic function, identify the following: the direction of opening the coordinates of the vertex the maximum or minimum value the equation of the axis of symmetry p = b 2a the x-intercepts and y-intercept the domain and range opens vertex: minimum value of y = when x = axis of symmetry: x = y-intercept occurs at and has a value of x-intercept occurs at and has a value(s) of domain: range: f(x) = ax 2 + bx + c q = c ap 2 c = y - intercept
11 For each graph of a quadratic function, identify the following: the direction of opening the coordinates of the vertex the maximum or minimum value the equation of the axis of symmetry p = b 2a the x-intercepts and y-intercept the domain and range opens vertex: minimum value of y = when x = axis of symmetry: x = y-intercept occurs at and has a value of x-intercept occurs at and has a value(s) of domain: range: f(x) = ax 2 + bx + c q = c ap 2 c = y - intercept
12 A frog sitting on a rock jumps into a pond. The height, h, in centimetres, of the frog above the surface of the water as a function of time, t, in seconds, since it jumped can be modelled by the function h(t) = -490t t Where appropriate, answer the following questions to the nearest tenth. a) Graph the function. b) What is the y-intercept? What does it represent in this situation? c) What maximum height does the frog reach? When does it reach that height? d) When does the frog hit the surface of the water? e) What are the domain and range in this situation? f) How high is the frog 0.25 s after it jumps? y = h(t) = height (h) x = time (t)
13 h(t) = -490t t y = f(x) = height (h) x = time (t) a) Graph the function.
14 h(t) = -490t t y = f(x) = height (h) x = time (t) a) Graph the function.
15 h(t) = -490t t y = f(x) = height (h) x = time (t) b) What is the y-intercept? What does it represent in this situation?
16 h(t) = -490t t y = f(x) = height (h) x = time (t) c) What maximum height does the frog reach? When does it reach that height?
17 h(t) = -490t t y = f(x) = height (h) x = time (t) d) When does the frog hit the surface of the water?
18 h(t) = -490t t y = f(x) = height (h) x = time (t) e) What are the domain and range in this situation?
19 h(t) = -490t t y = f(x) = height (h) x = time (t) f) How high is the frog 0.25 s after it jumps?
20 A diver jumps from a 3-m springboard with an initial vertical velocity of 6.8 m/s. Her height, h, in metres, above the water t seconds after leaving the diving board can be modelled by the function h(t) = -4.9t t + 3. a) Graph the function. b) What is the y-intercept? What does it represent in this situation? c) What maximum height does the diver reach? When does it reach that height? d) How long does it take before the diver hits the water? e) What are the domain and range in this situation? f) What is the height of the diver 0.6 s after leaving the board? y = f(x) = height (h) x = time (t)
21 h(t) = -4.9t t + 3 y = f(x) = height (h) x = time (t) a) Graph the function.
22 h(t) = -4.9t t + 3 y = f(x) = height (h) x = time (t) a) Graph the function.
23 h(t) = -4.9t t + 3 y = f(x) = height (h) x = time (t) b) What is the y-intercept? What does it represent in this situation?
24 h(t) = -4.9t t + 3 y = f(x) = height (h) x = time (t) c) What maximum height does the diver reach? When does it reach that height?
25 h(t) = -4.9t t + 3 y = f(x) = height (h) x = time (t) d) How long does it take before the diver hits the water?
26 h(t) = -4.9t t + 3 y = f(x) = height (h) x = time (t) e) What are the domain and range in this situation?
27 h(t) = -4.9t t + 3 y = f(x) = height (h) x = time (t) f) What is the height of the diver 0.6 s after leaving the board?
28 3.2 HOMEWORK (part 1) OPages: 174 OProblems: 2, 5, 7, 12
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34 A rancher has 100 m of fencing available to build a rectangular corral. a) Write a quadratic function in standard form to represent the area of the corral. b) What are the coordinates of the vertex? What does the vertex represent in this situation? c) Sketch the graph for the function you determined in part a). d) Determine the domain and range for this situation. e) Identify any assumptions you made in modelling this situation mathematically.
35 A rancher has 100 m of fencing available to build a rectangular corral. a) Write a quadratic function in standard form to represent the area of the corral.
36 A rancher has 100 m of fencing available to build a rectangular corral. b) What are the coordinates of the vertex? What does the vertex represent in this situation?
37 A rancher has 100 m of fencing available to build a rectangular corral. c) Sketch the graph for the function you determined in part a).
38 A rancher has 100 m of fencing available to build a rectangular corral. c) Sketch the graph for the function you determined in part a).
39 A rancher has 100 m of fencing available to build a rectangular corral. c) Sketch the graph for the function you determined in part a).
40 A rancher has 100 m of fencing available to build a rectangular corral. d) Determine the domain and range for this situation.
41 A rancher has 100 m of fencing available to build a rectangular corral. e) Identify any assumptions you made in modelling this situation mathematically.
42 At a children s music festival, the organizers are roping off a rectangular area for stroller parking. There is 160 m of rope available to create the perimeter. a) Write a quadratic function in standard form to represent the area of the stroller parking. b) What are the coordinates of the vertex? What does the vertex represent in this situation? c) Sketch the graph for the function you determined in part a). d) Determine the domain and range for this situation. e) Identify any assumptions you made in modelling this situation mathematically.
43 At a children s music festival, the organizers are roping off a rectangular area for stroller parking. There is 160 m of rope available to create the perimeter. a) Write a quadratic function in standard form to represent the area of the stroller parking.
44 At a children s music festival, the organizers are roping off a rectangular area for stroller parking. There is 160 m of rope available to create the perimeter. b) What are the coordinates of the vertex? What does the vertex represent in this situation?
45 At a children s music festival, the organizers are roping off a rectangular area for stroller parking. There is 160 m of rope available to create the perimeter. c) Sketch the graph for the function you determined in part a).
46 At a children s music festival, the organizers are roping off a rectangular area for stroller parking. There is 160 m of rope available to create the perimeter. d) Determine the domain and range for this situation.
47 At a children s music festival, the organizers are roping off a rectangular area for stroller parking. There is 160 m of rope available to create the perimeter. e) Identify any assumptions you made in modelling this situation mathematically.
48 3.2 HOMEWORK (part 2) OPages: OProblems: 15, 17, 19
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