Algebra 5.9

1. Algebra 5.9 Transforming Linear Functions

2. Learning Targets Language Goal • Students will be able to describe how changing slope and y-intercept affect the graph of a linear function. Math Goal • Students will be able to explain the affects of changing portions of a linear function. Essential Question • What information can you get from a graph?

3. Warm-up

4. Homework Check

5. Homework Check

6. Homework Check

7. Homework Check

8. Vocabulary • Family of Functions: • Parent Function: • Transformations: Set of functions whose graphs have basic characteristics in common. The most basic function in a family for linear functions f(x) = x A change in position or size of a figure

9. Vocabulary • 3 Types of Transformations • Translation: • Rotation: • Reflection: A transformation that moves every point the same distance in the same direction. A transformation that “turns” about a point A transformation across a line that produces a mirror image or “flip”

10. Vertical Translation of a Linear Function • When the y-intercept b is changed in the function the graph is translated vertically. • If b increases, the graph translates up • If b decreases, the graph translates down.

11. Example 1: Translating Linear Functions • Graph the two equations. Then describe the transformation form the graph of f(x) to g(x). A. and Translates 5 units down.

12. Example 1: Translating Linear Functions • Graph the two equations. Then describe the transformation form the graph of f(x) to g(x). B. and Translates 6 units down.

13. Example 1: Translating Linear Functions • Graph the two equations. Then describe the transformation form the graph of f(x) to g(x). C. and Translates 6 units up.

14. Rotation of a Linear Function • When the slope m is changed in the function it causes a rotation of the graph about the point (0, b). Which changes the line’s steepness. • If m increases, the line gets steeper • If m decreases, the line gets flatter.

15. Example 2: Rotating Linear Functions • Graph the two equations. Then describe the transformation form the graph of f(x) to g(x). A. and g(x) is steeper than f(x). Rotate graph about point (0, 2)

16. Example 2: Rotating Linear Functions • Graph the two equations. Then describe the transformation form the graph of f(x) to g(x). B. and g(x) is flatter than f(x). Rotate graph about point (0, -1)

17. Example 2: Rotating Linear Functions • Graph the two equations. Then describe the transformation form the graph of f(x) to g(x). C. and g(x) is steeper than f(x). Rotate graph about point (0, -0)

18. Reflection of a Linear Function • When the slope m is multiplied by -1 in , the graph is reflected across the y – axis.

19. Example 3: Reflecting Linear Functions • Graph f(x). Then reflect the graph of f(x) across the y – axis. Write a function g(x) to describe the new graph. A. Step 1: Find m. Step 2: Multiply m by -1. Step 3: Re-graph the new line. Equation of Reflection:

20. Example 3: Reflecting Linear Functions • Graph f(x). Then reflect the graph of f(x) across the y – axis. Write a function g(x) to describe the new graph. B. Step 1: Find m. Step 2: Multiply m by -1. Step 3: Re-graph the new line. Equation of Reflection:

21. Example 3: Reflecting Linear Functions • Graph f(x). Then reflect the graph of f(x) across the y – axis. Write a function g(x) to describe the new graph. C. Step 1: Find m. Step 2: Multiply m by -1. Step 3: Re-graph the new line. Equation of Reflection:

22. Example 4: Multiple Transformations of Linear Functions • If needed graph f(x) and g(x). Then describe the transformation from f(x) to g(x). • A. and f(x) slope = 1 g(x) slope = 3 This rotates the graph about (0, 0) and makes it steeper. f(x) b = 0 g(x) b = 1 This translates the graph up 1 unit.

23. Example 4: Multiple Transformations of Linear Functions • If needed graph f(x) and g(x). Then describe the transformation from f(x) to g(x). • B. and f(x) slope = 1 g(x) slope = -1 This reflects the graph across the y – axis f(x) b = 0 g(x) b = 2 This translates the graph up 2 units.

24. Example 4: Multiple Transformations of Linear Functions • If needed graph f(x) and g(x). Then describe the transformation from f(x) to g(x). • C. and f(x) slope = 1 g(x) slope = 2 This rotates the graph about (0, 0) and makes it steeper. f(x) b = 0 g(x) b = – 3 This translates the graph down 3 units.

25. Example 4: Multiple Transformations of Linear Functions • If needed graph f(x) and g(x). Then describe the transformation from f(x) to g(x). • D. and f(x) slope = – 1 g(x) slope = – 4 This rotates the graph about (0, 0) and makes it steeper. f(x) b = –1 g(x) b = 0 This translates the graph up 1 unit.

26. Example 4: Multiple Transformations of Linear Functions • If needed graph f(x) and g(x). Then describe the transformation from f(x) to g(x). • E. and f(x) slope = – 2 g(x) slope = 3 This rotates the graph about (0, 0) and makes it steeper. Since the slope goes from negative to positive we know the graph is reflected across the y – axis. f(x) b = 2 g(x) b = – 1 This translates the graph down 3 units.

27. Example 5: Word Problems A. A trophy company charges $175 for a trophy plus $0.20 per letter for the engraving. The total charge for the trophy with x letters is given by the function f(x) = 0.20x + 175. How will the graph change if the trophy’s cost is lowered to $172? If the charge per letter is raised to $0.50?

28. Example 5: Word Problems B. How will the graph change if the charger per letter is lowered to $0.15? IF the trophy’s cost is raised to $180?

29. Lesson Quiz