Section 6.2 Properties of Graphs of Quadratic Functions soln.notebook January 12, 2017
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1 Section 6.2: Properties of Graphs of Quadratic Functions 1
2 Properties of Graphs of Quadratic Functions A quadratic equation can be written in three different ways. Each version of the equation gives information about the graph of the parabola. The three equations are: 1) f(x) = ax 2 + bx + c Standard Form 2) f(x) = a(x r)(x s) Factored Form 3) f(x) = a(x h) 2 + k Vertex Form 2
3 Reminders The vertex of a parabola is the maximum or minimum point. > How to find: The parabola can be cut perfectly in half about a line called the axis of symmetry.the equation of the axis of symmetry (AOS) is x = a (the x coordinate of the vertex) > How to find: 3
4 Find the vertex and axis of symmetry for each given parabola. 1. y = x 2 8x +7 x coordinate: Axis of symmetry: y coordinate: Vertex 2. y= 2x 2 8x 15 x coordinate: Axis of symmetry: y coordinate: vertex 4
5 Determine the vertex of the following. 1. Does it have a maximum or minimum point? The vertex is. 2. Does it have a maximum or minimum point? The vertex is. 5
6 3. Given the equation y = 2x 2 8x + 7 Does it have a maximum or minimum point? The vertex is. 6
7 Example. Consider the following function: i. Does this function have a minimum or maximum point? ii. What is the minimum or maximum point? iii.what is the minimum or maximum value? iv.what do you know about the coefficient "a" in the equation of this quadratic? v. What is the equation of the axis of symmetry? vi.what is the vertex of this quadratic function? 7
8 Study the following graphs and make a conjecture connecting the x intercepts and the axis of symmetry: 1. The x intercepts are: The vertex is: 8
9 The axis of symmetry can also be linked to the x intercepts in the table of values of a quadratic function. Consider the following table of values: 1. How are the x intercepts determined using a table of values? 2. Can you identify the vertex from the table? 9
10 Domain and Range of a Quadratic Function Recall: Domain is the set of all input values (or x values). The domain is independent. NOTE: For the content of this course, the domain will ALWAYS be Range is the set of all output values (or y values). The range is dependent and is determined by the domain ( x variable). 10
11 1. Determining Domain & Range Graphically Consider the following graphs: Vertex: A. Direction of Opening: Minimum/Maximum Value: Domain: Range: B. Vertex: Direction of Opening: Minimum/Maximum Value: Domain: Range: 11
12 You try!! C. Vertex: Direction of Opening: Minimum/Maximum Value: Domain: Range: D. Vertex: Direction of Opening: Minimum/Maximum Value: Domain: Range: 12
13 2. Determining Domain and Range Algebraically (Using a Function) How do we attain the domain of a quadratic function such as y = 2x 2 + 4x + 1 without the aid of a graph? A. What is the direction of opening for the given function? B. Will the function have a maximum or minimum value? C. How can we algebraically attain the maximum/minimum value? D. How does the above information enable us to express the range? 13
14 Determine the vertex, domain, and range for the following functions: A. y = 3x 2 18x + 5 Vertex: Domain: Range: 14
15 Summary: To attain the domain and range Domain: For any quadratic function is Range: (i) Determine the of opening (ii) Determine the of vertex by (iii) Substitute the result from (ii) into the function to get the maximum/minimum value, which is the of the vertex (iv) State the range. If a > 0, then If a < 0, then 15
16 Graph the following functions: A. y = x 2 2x + 3 Direction of Opening: Vertex: Table of Values: x y Domain: Range: y intercept: Number of x intercepts: NOTE: How many x-intercepts are possible?? 16
17 In a game of football, a team can score by kicking the ball over a bar and between two uprights. For a kick in a particular game, the height of the ball above the ground, h(t), in meters can be modelled by the function: where t is the time in seconds after the ball left the foot of the player. a) What is the initial height of the ball b) Determine the axis of symmetry. What does this value represent? c) Determine the vertex. What does this represent in the context of the question. d) Determine the value of h(1.5). What does this value represent? 17
18 e) If the bar that the kicker is trying to get the ball over is 30 m high, does the team score a point? f) Assuming the ball hits the ground at 5.2 seconds, sketch the graph of the function. 18
19 19
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