Objectives

1. Objectives • Vertical Shifts Up and Down • Horizontal Shifts to the Right and Left • Reflections of Graphs • Stretching and Shrinking of Graphs

2. Laws for Graphing Shifts • All graphs must be placed on graphing paper. We do not use ruled paper for graphing. • Graphs are to be drawn using a straight edge. • Arrows are to be placed on each end of the x and y-axis. • Label the x and y-axis. • When graphing two functions on the same graph, use different colored pencils or markers. • Label points. • Be sure the equation(s) is located on the graph. • Keep everything color-coded. • Use a straight-edge where needed. Straight lines must not be sketched. • Curved lines are sketched to the best of your ability. • Be sure your graph is very neat and professional.

3. Parent Function • What is a parent function? • A parent function is the most basic form of a function. It has the same basic properties as other functions like it, but it has not been transformed in any way. • Parent functions allow us to quickly tell certain traits of their “children” (which have been transformed).

4. Parent Functions The six most commonly used parent functions are: • Constant Function • Linear (Identity) Function • Absolute Value Function • Square root Function • Quadratic Function • Cubic Function

5. Constant Function f(x) = c wherec is a constant   Characteristics: f(x)=a is a horizontal line. It is increasing and continuous on its entire domain (-∞, ∞).

6. Linear Function (Identity)  f(x) = x Characteristics: f(x)=x is increasing and continuous on its entire domain (-∞, ∞). 

7. Absolute Value Function  f(x) = │x │  Characteristics: is a piecewise function. It decreases on the interval (-∞, 0) and increases on the interval (0, ∞). It is continuous on its entire domain (- ∞, ∞). The vertex of the function is (0, 0).

8. Square Root Function f(x) = x   Characteristics: f(x)=√x increases and is continuous on its entire domain [0, ∞). Note: x≥0 for f to be real.

9. Quadratic Function f(x) = x 2 Characteristics: f(x)=x2is continuous on its entire domain (-∞, ∞). It is increasing on the interval (0, ∞) and decreasing on the interval (-∞, 0). Its graph is called a parabola, and the point where it changes from decreasing to increasing, (0,0), is called the vertex of the graph.

10. Cubic Function  f(x) = x 3 Characteristics: f(x)=x3increases and is continuous on its entire domain (-∞, ∞). The point at which the graph changes from “opening downward” to “opening upward” (the point (0,0)) is called the origin.)   

11. Vertical Shifts The graph represents the equation f(x) We add or subtract to the output of the function. Translates the function vertically (+ up, - down)

12. Vertical Shifts   Graph         What is the shift of the graph

13. Vertical Shifts

14. y 8 4 x 4 -4 -4 Example: Use the graph of f(x) = |x| to graph thefunctions g(x) = |x| +3 and h(x) = |x| –4. g(x) = |x| + 3 f(x) = |x| h(x) = |x| –4

15. y x VerticalShifts If c is a positive real number, the graph of f(x) + cis the graph of y = f(x) shifted upwardc units. If c is a positive real number, the graph of f(x) –cis the graph of y = f(x) shifted downwardc units. f(x) + c f(x) +c f(x) –c -c

16. Graphing Neatly graph your parent function (with a colored pencil). Plot some points for reference. Shift each point from your parent function up c units (if c is positive) or down c units (if c is negative). Use a different colored pencil.

17. Horizontal Shifts Start with f(x) The c is now in parentheses with x. Add or subtract to the input of the function. Translates the function horizontally (+ left, - right)

18. Horizontal Shifts Graph               What is the shift of the graph ?

19. Horizontal Shifts

20. Horizontal Shifts

21. y 4 4 -4 x Example: Use the graph of f(x) = x3to graphg(x) = (x – 2)3 and h(x) = (x + 4)3. f(x) = x3 g(x) = (x – 2)3 h(x) = (x + 4)3

22. y x Horizontal Shifts If c is a positive real number, then the graph of f(x – c) is the graph of y = f(x) shifted to the rightc units. -c +c If c is a positive real number, then the graph of f(x + c) is the graph of y = f(x) shifted to the leftc units. y = f(x + c) y = f(x) y = f(x – c)

23. Graphing Horizontal shifts don’t make sense when looking at the equation. Everything is in parentheses. Think opposite direction in your shift. Neatly graph your parent function (with a colored pencil). Plot some points for reference. Shift each point from your parent function to the left cunits (if c is positive) or to the right cunits (if c is negative). Use a different colored pencil.

24. Reflections about the x-axis What do we know about the x-coordinates in these two graphs?  They are the same. What do we know about the y-coordinates in these two graphs?  They are the opposite.

25. When the right side of a function is multiplied by , the graph of the new function is the reflection about the x-axis of the graph of the function . The x-axis acts as a mirror of these two functions.

26. Reflections about the y-axis When x is replaced with , the graph of the new function is the reflection about the y-axis. The y-axis acts as a mirror of the two functions.

28. What do we observe about the y-coordinates? What do we observe about the x-coordinates? (9, 3) (-9, 3) (-4, 2) (4, 2) (-1, 1) (1, 1)

29. Vertical Stretching of Graphs The x-coordinates stay the same. The y-coordinates increase by a factor of 2. 2 is the constant c. Our graph stretches vertically. That is, it moves away from the x-axis to become more vertical.

30. Vertical Shrinking of Graphs The x-coordinates stay the same. The y-coordinates decrease by a factor of ½. ½ is the constant c. Our graph shrinks vertically. That is, it moves more towards the x-axis to be less vertical.

31. y 4 is the graphof y = x2shrunk vertically bya factor of . x –4 4 Vertical Stretching and Shrinking If c > 1 then the graph of y= c f(x) is the graph of y = f(x) stretched vertically by c. (Remember: c is the constant) If 0 < c < 1 then the graph of y = c f(x) is the graph of y = f(x) shrunk vertically by c. y = x2 y =2x2 Example: y =2x2 is the graph of y = x2stretched verticallyby a factor of 2.

32. Horizontal Stretching of Graphs ½ is the constant, c. The x-coordinates stay the same. The y-coordinates decrease by a factor of ½, which is the same as . Our graph stretches horizontally. When the constant is greater than zero and less than one, the graph is stretching horizontally.

33. Horizontal Shrinking of Graphs 2 is the constant, c. The x-coordinates stay the same. The y-coordinates increase by a factor of 2, which is the same as . Our graph shrinks horizontally. When the constant is greater than one, the graph is shrinking horizontally.

34. y 4 is the graph of y = |x| stretched horizontallyby 2 . x -4 4 Horizontal Stretching and Shrinking If c > 1, the graph of y = f(cx) is the graph of y = f(x) shrunk horizontally by 1/c. If 0 < c < 1, the graph of y = f(cx) is the graph of y = f(x) stretchedhorizontally by 1/c. y = |2x| Example:y = |2x|is the graph of y = |x| shrunk horizontally by 1/2. y = |x|

35. Graphing a New Function from the Original Function

36. Homework: Pages 216 – 217 1, 3, 7, 17, 19, 27, 53, 55, 67, 69, 81, 83, 87, 95, 97