3.1 Quadratic Functions in Vertex Form

Transcription

1 3.1 Quadratic Functions in Vertex Form 1) Identify quadratic functions in vertex form. 2) Determine the effect of a, p, and q on the graph of a quadratic function in vertex form where y = a(x p)² + q 3) Analyse and graph quadratic functions using transformations. 1

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16 y = a(x p) 2 + q "a" "q" "p" a > 0, opens up a < 0, opens down a > 1, narrow a < 1, wide a = 1, regular q > 0, vert. shift up q < 0, vert. shift down p > 0, horz. shift right p < 0, horz. shift left 16

17 Drag the equation to the matching graph y = x 2 y = x 2 3 y = 3x 2 y = x y = (x 3) 2 y = (x + 3) 2 Click for answer 17

18 Drag the vertex to the matching graph (0, 3) (5, 4) ( 4, 2) (2, 4) (0, 0) ( 3, 0) (4, 5) ( 4, 2) (2, 4) (3, 0) Click for answer 18

19 Drag the vertex to the matching equation (0, 0) (0, 3) ( 3, 0) y = ¼x 2 y = 2x 2 3 y = (x + 3) 2 y = (x + 4) 2 2 ( 4, 2) y = (x 5) (5, 4) y = (x 3) 2 (3, 0) (2, 4) (4, 5) ( 4, 2) (2, 4) Click for answer 19

20 3.1 Quadratic Functions Vertex Form Lesson Focus: Sketch Graphs of Quadratic Functions Example 1: Sketch Graphs of Quadratic Functions in Vertex Form Determine the following characteristics for each function. the vertex the domain and range the direction of opening the equation of the axis of symmetry Then, sketch each graph. a) y = 2(x + 1) 2 3 b) y = 0.25(x 4) Quadratic Functions Vertex Form 20

21 3.1 Quadratic Functions Vertex Form Lesson Focus: Sketching Quadratic Functions Your Turn Determine the following characteristics for each function. the vertex the domain and range the direction of opening the equations of the axis of symmetry Then, sketch each graph. a) y = (x 2) 2 4 b) y = 3(x + 1) Answer Part a Answer Part b 3.1 Quadratic Functions Vertex Form 21

22 3.1 Quadratic Functions Vertex Form Lesson Focus: Equation of Quadratic Functions Example 2: Determine a Quadratic Function in Vertex Form Given Its Graph a) Instructions 3.1 Quadratic Functions Vertex Form 22

23 3.1 Quadratic Functions Vertex Form Lesson Focus: Equation of Quadratic Functions Example 2: Determine a Quadratic Function in Vertex Form Given Its Graph b) 3.1 Quadratic Functions Vertex Form 23

24 3.1 Quadratic Functions Vertex Form Lesson Focus: Equation of Quadratic Functions Your Turn Determine a quadratic function in vertex form for each graph. a) b) 3.1 Quadratic Functions Vertex Form 24

25 What is an x intercept? What possibilities exist for x intercepts with regards to quadratic functions? 25

26 Sketch each quadratic function and determine the number of x intercepts for each function. 26

27 Without graphing, determine the number of x intercepts for each quadratic function. 27

28 The deck of the Lions' Gate Bridge in Vancouver is suspended from two main cables attached to the tops of two supporting towers. Between the towers, the main cables take the shape of a parabola as they support the weight of the deck. The towers are 111 m tall relative to the water's surface and are 472 m apart. The lowest point of the cables is approximately 67 m above the water's surface. a) Model the shape of the cables with a quadratic function in vertex form. b) Determine the height above the surface of the water of a point on the cables that is 90 m horizontally from one of the towers. Express your answer to the nearest tenth of a metre. 28

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30 The towers are 111 m tall relative to the water's surface and are 472 m apart. The lowest point of the cables is approximately 67 m above the water's surface. b) Determine the height above the surface of the water of a point on the cables that is 90 m horizontally from one of the towers. Express your answer to the nearest tenth of a metre. 30

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33 The towers are 111 m tall relative to the water's surface and are 472 m apart. The lowest point of the cables is approximately 67 m above the water's surface. a) Model the shape of the cables with a quadratic function in vertex form. 33

34 The towers are 111 m tall relative to the water's surface and are 472 m apart. The lowest point of the cables is approximately 67 m above the water's surface. b) Determine the height above the surface of the water of a point on the cables that is 90 m horizontally from one of the towers. Express your answer to the nearest tenth of a metre. 34

35 Suppose a parabolic archway has a width of 280 cm and a height of 216 cm at its highest point above the floor. a) Write a quadratic function in vertex form that models the shape of this archway. b) Determine the height of the archway at a point that is 50 cm from its outer edge. 35

36 The path of a rocket is described by the function where h(t) is the height of the rocket, in metres, and t is the time, in seconds, after the rocket is fired. a) What is the maximum height reached by the rocket? 36

37 The path of a rocket is described by the function where h(t) is the height of the rocket, in metres, and t is the time, in seconds, after the rocket is fired. b) How many seconds after it was fired did the rocket reach this height? 37

38 c) How high above the ground was the rocket when it was fired? 38

39 The vertex of a parabola is ( 2, 4). One x intercept is 7. What is the other x intercept? 39

40 The x intercepts of a parabola are 5, and 7. What is the equation of the axis of symmetry? 40

41 Pgs: : # s 1, 2ab, 3bc, 4 7, 8ab, 9 15, 20, 21 41