1.2 Reflections and Stretches

Transcription

1 Chapter Part : Reflections.2 Reflections and Stretches Pages 6 3 Investigating a reflection in the x axis:. a) Complete the following table for and sketch on the axis provided. x y

2 b) Now sketch the reflection in the x axis by using the x axis as a mirror and complete the following table using your results. What do you notice? x y Conclusion: Mapping Notation: Equation: invariant point(s) a point or points that remain unchanged after a transformation has been applied to it. For reflections, the invariant point(s) lie(s) on the line of reflection. 2

3 Investigating a reflection in the y axis: 2. a) Complete the following table for and sketch on the axis provided. x y 3

4 b) Sketch the reflection in the y axis by using it as a mirror and complete the following table of values. What do you notice? x y Conclusion: Mapping Notation: Equation: 4

5 Generally: If the function y = f(x) is transformed to give y = f(x), If the function y = f(x) is transformed to give y = f( x), 5

6 Example Page 8 Compare the Graphs of y = f(x), y = f(x), and y = f( x) a) Given the graph of y = f(x), graph the functions y = f(x) and y = f( x). b) How are the graphs of y = f(x) and y = f( x) related to the graph of y = f(x)? 6

7 Example : Your Turn Page 20 a) Given the graph of y = f(x), graph the functions y = f(x) and y = f( x). b) Show the mapping of key points on the graph of y = f(x) to image points on the graphs of y = f(x) and y = f( x). c) Describe how the graphs of y = f(x) and y = f( x) are related to the graph of y = f(x). State any invariant points. Answer 7

8 Part 2: Stretches Investigating a stretch about the x axis:. a) Complete the following table for and sketch on the axis provided. x y

9 b) Sketch the following using the graphing calculator and use the table feature to complete the given table of values. Then sketch on the axis provided. () (2) x y x y 9

10 c) Complete the following table. Function Describe how the table changed Mapping Notation Domain Range () (2) 0

11 d) Sketch (2) and (3) using the graphing calculator and use the table feature to complete the given table of values. Then sketch on the axis provided. () (2) (3) x y x y x y

12 2

13 e) Complete the following table. Function Describe how the table changed Mapping Notation Domain Range (2) (3) 3

14 Generally: If the function y = f(x) is transformed to give y = af(x), where a > 0(if a is negative, there is a reflection in the x axis), Mapping Notation: If the function y = f(x) is transformed to give y = f(bx), where b > 0 (if b is negative, there is a reflection in the y axis), Mapping Notation: 4

15 Example 2 Page 2 Graph y = af(x) Given the graph of y = f(x), transform the graph of f(x) to sketch the graph of g(x) describe the transformation state any invariant points state the domain and range of the functions a) g(x) = 2f(x) b) 5

16 Example 2: Your Turn Page 22 Given the function f(x) = x 2, transform the graph of f(x) to sketch the graph of g(x) describe the transformation state any invariant points state the domain and range of the functions a) g(x) = 4f(x) Answer a) b b) g(x) = f(x) Answer b) 6

17 Example 3 Page 23 Graph y = f(bx) Given the graph of y = f(x), transform the graph of f(x) to sketch the graph of g(x) describe the transformation state any invariant points state the domain and range of the functions a) g(x) = f(2x) b) g(x) = f( x) 2 7

18 Example 3: Your Turn Page 24 Given the function f(x) = x 2, transform the graph of f(x) to sketch the graph of g(x) describe the transformation state any invariant points state the domain and range of the functions a) g(x) = f(3x) b) g(x) = f( x) 4 Answer a) Answer b) 8

19 Example 4 Page 25 Write the Equation of a Transformed Function The graph of the function y = f(x) has been transformed by either a stretch or a reflection. Write the equation of the transformed graph, g(x). 9

20 Example 4: Your Turn Page 27 The graph of the function y = f(x) has been transformed. Write the equation of the transformed graph, g(x). Answer 20

21 Page 27 2