Properties of Graphs of Quadratic Functions
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1 H e i g h t (f t ) Lesson 2 Goal: Properties of Graphs of Quadratic Functions Identify the characteristics of graphs of quadratic functions: Vertex Intercepts Domain and Range Axis of Symmetry and use the graphs to solve problems. (I) Using symmetry to estimate the coordinates of a point The height of a volleyball against time (i) 10 At what time did the volleyball reach its greatest height? h Time (s) t
2 (ii) Use the graph to approximate the greatest height? (iii) Using the answers to (i) and (ii), what are the coordinates of the highest point? Vertex: The point at which the quadratic function reaches its maximum or minimum value. Maximum Value When the graph opens down, the vertex is the point on the graph and the y-coordinate of the vertex is the value. Minimum Value When the graph opens up the vertex is the point on the graph and the y-coordinate is the value. The value of a in a quadratic function y = ax 2 + bx + c determines whether the vertex is a maximum or minimum point.
3 H e i g h t (f t ) (II) Attaining a maximum or minimum value through a table of values or graph A golf ball is struck and its height with respect to time is represented by the function h(t) = 3t t where h(t) represents height and t is the time in seconds. (c) (d) (e) What s the direction of opening? Will the ball attain a maximum or minimum height? What s the y intercept? Create a table of values and graph the function. t h(t) How long does it take for the ball to attain its maximum height? h Time (s) t (f) What is the maximum height attained? (g) What are the coordinates of the vertex?
4 (III) Attaining a maximum or minimum value through a graph and formula For each function indicated in the table determine: (i) (ii) (iii) (iv) the vertex (using graphing software the equation of axis of symmetry the values of a, b and c from the function y = ax 2 + bx + c the value of Function y = ax 2 + bx + c Vertex Equation of Axis of Symmetry a b c y = x 2 4x + 7 y = 2x 2 4x + 7 y = 3x 2 6x + 10 What do you notice about the x coordinate of each vertex, the equation of axis of symmetry and the value of? Once the x coordinate of the vertex is attained from a quadratic function such as y = 2x 2 4x + 7, how could we algebraically attain the y coordinate?
5 Summary Attaining the vertex of a quadratic function y = ax 2 + bx + c (i) Get the x coordinate of the vertex by the formula x = (ii) Substitute that result into y = ax 2 + bx + c to attain the y coordinate. The vertex and equation of axis of symmetry of a quadratic function can be attained: (A) Graphically State the vertex and equation of axis of symmetry. Vertex: Equation of axis of symmetry: (B) Tabulation State the vertex and equation of axis of symmetry. Vertex: Equation of axis of symmetry:
6 (C) Algebraically Determine the vertex and equation of axis of symmetry for y = 2x 2 8x + 7. (IV) Determining the axis of symmetry from a set of points. Determine the equation of axis of symmetry from the parabola. Where is the line of symmetry positioned compared to the location of the two given points?
7 Summary Axis of symmetry (i) A vertical reflection line that passes through the vertex (ii) Can be attained by the formula x = when the quadratic function y = ax 2 + bx + c is given. (iii) Can be attained from two points with the same y coordinate by averaging the x coordinates. Determine the equation of axis of symmetry for each parabola that contains the points: ( 2, 4) and (6, 4) (5, 0) and (11, 0) (V) Attaining the domain and range of a quadratic function Review of Domain and Range Domain is the set of all input values (or x values) Range is the set of all output values (or y values) The domain and range can be attained: (i) Graphically (ii) Tabulation (or set of points) (iii) Function
8 Determining Domain & Range Graphically 1. State the domain using set notation for: (c) (d) State the range using set notation for:
9 3. State the domain and range for: NOTE: For Domain - all points on the graph FALL to the x-axis For Range - all points on the graph MOVE OVER to the y-axis y y x x Domain: Domain: Range: Range:
10 Determining Domain & Range from a Quadratic Function How do we attain the domain of a quadratic function such as y = 2x 2 + 4x + 1 without the aid of a graph? What is the direction of opening for the given function? 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? SUMMARY: To attain the domain and range from y = ax 2 + bx + c Domain for any quadratic function is Range (i) determine the of opening (ii) (iii) determine the of vertex by Substitute the result from (ii) into the function to get the maximum/minimum value (iv) State the range.
11 Determine the domain and range for: y = 3x 2 2 y = x 2 + 4x + 4 (c) y = x 2 + 6x 8 P #2 (state the y intercept), #4, #5, #6, #7a(i), b, c, #9 b, c, #10, #11b, c #13a, b #14
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