3 3.2 Investigating Quadratic Functions in Standard Form

Transcription

1 Chapter Investigating Quadratic Functions in Standard Form Focus On... identifying quadratic functions in standard form determining the vertex, domain and range, axis of symmetry, maximum or minimum value, and x intercepts and y intercept for quadratic functions in standard form graphing and analysing quadratic functions in applied situations Identify Characteristics of a Quadratic Function in Vertex Form 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 the x intercepts and y intercept the domain and range Mar 27 11:52 AM 1

2 Repeat for the following. Mar 27 11:54 AM 3.2 Example 1: Your Turn For each 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 the x intercepts and y intercept the domain and range a) y = x 2 + 6x + 5 b) y = x 2 + 2x + 3 Answer Part a) Answer Part b) 2

3 Can you complete either of the problems below (same instructions)? or Mar 27 11:55 AM We are able to have a good understanding of the parabola if the equation is written in vertex form. Vertex (p, q) Equation of axis of symmetry x = p Domain xer If a>0 then the parabola opens upwards, the min = q and the range is. If a<0 then the parabola opens downwards, the max = q and the range is. Mar 27 12:51 PM 3

4 Expand y=5(x 3) 2 +6 to standard form seen below. Apr 2 9:49 AM Expand form to write it standard rather than vertex form. Mar 27 1:01 PM 4

5 Our new formulas will help us when the quadratic equation is in standard form rather than vertex form. You can use them to switch from standard form to vertex form. Mar 27 1:05 PM Write the following in standard form. If it is a quadratic, find the vertex. Apr 2 9:55 AM 5

6 Given the equation find the vertex, equation of symmetry, max/min, domain, range and sketch the parabola. Also write the equation in vertex form. y=3x 2 +6x+11 Apr 2 9:50 AM Apr 2 9:53 AM 6

7 3.2 Quadratic Functions Standard Form Lesson Focus: Equation of Quadratic Functions Example 1d: Identify Characteristics of a Quadratic Function in Standard Form direction of opening coordinates of vertex max/min equation for axis of symmetry x intercept(s) and y intercept domain and range 3.2 Quadratic Functions Standard Form example 1d Oct 28 9:08 AM 7

8 3.2 Example 2 Analysing a Quadratic Function 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? (Use technology) e) What are the domain and range in this situation? f) How high is the frog 0.25 s after it jumps? Continue Next Page 3.2 Example 2: Your Turn 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 does the y intercept represent? c) What maximum height does the diver reach? When does she reach that height? d) How long does it take before the diver hits the water? e) What domain and range are appropriate in this situation? f) What is the height of the diver 0.6 s after leaving the board? Answer 8

9 3.2 Example 3 Write a Quadratic Function to Model a Situation 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. Continue Next Page 3.2 Example 3: Your Turn 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 for 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. Answer 9

10 Oct 17 2:59 PM 10