1. Chapter 3.4 Graphs and Transformations By Saho Takahashi

2. Parent Functions Parent Function – is a function with a certain shape that has the simplest algebraic rule for that shape. The parent functions are used to illustrate the rules for the basic transformations.

3. Parent Functions

6. Vertical Shifts Let c as a positive number. The graph of g(x) = f(x) + c is the graph of f shifted upward c units. The graph of g(x) = f(x) – c is the graph of f shifted downward c units.

7. Example 1 - Shifting a graph Vertically Q. Graph g(x) = |x|+ 4, and h(x) = |x|- 3 Answer : The parent function of g(x) and h(x) is f(x) = |x|. The graph of g(x) is the graph of f(x) = |x| shifted 4 units upward, and the graph of h(x) is the graph of f(x) = |x| shifted 3 units downward.

8. Horizontal Shifts Let c be a positive number. The graph of g(x)= f(x + c) is the graph of f shifted c units to the left. The graph of g(x) = f(x - c) is the graph of f shifted c units right.

9. Example 2 – Shifting a Graph Horizontally Q. Graph g(x) = 1/ x – 3 and h(x) = 1/ x + 4. Answer : the parent function of g(x) and h(x) is f(x) = 1/ x. The graph of g(x) is the graph of f(x) = 1/ x shifted 3 units to the right, and the graph of h(x) = 1 / x shifted 4 units to the left.

10. Reflections The graph of g(x) = - f(x) is the graph of f reflected across the x – axis. The graph of g(x) = f( -x ) is the graph of f reflected across the y – axis.

11. Example 3 – Reflecting a graph Across the x – or y – axis. Graph g(x) = - [x] and h(x) = [-x] Answer: The parent function of functions g(x) and h(x) is f(x) = [x]. the graph of g(x) is the graph of f(x) = [x] reflected across the x – axis, and the graph of h(x) is the graph of f(x) = [x] reflected across the y – axis.

12. Compression and Stretches Graphs of function may be either compressed or stretched vertically and horizontally.

13. Vertically Stretches and Compressions In the function of the graph g(x) =c· f(x) c > 0…. If c > 1, the graph of g(x) = c· f(x) is the graph of f stretched vertically, away from the x – axis, by a factor of c. If c < 1, the graph of g(x) = c· f(x) is the graph of f compressed vertically, toward the x – axis, by a factor of c.

14. Example 4 Q. Graph g(x) =2³and h(x) =1/4x³ Answer: The parent function is f(x) = x³. For the graph of g(x), every y coordinate of the parent function is multiplied by 2, stretching the graph of the function in the vertical direction away from the x axis. For the function h(x)m every y – coordinate of the parent function is multiplied by ¼, compressing the graph of the function in vertical direction toward the x – axis.

15. Horizontal Stretches and Compressions In the function of the graph of g(x) = f( c· x ). If c > 1, the graph of g(x) = f( c· x ) is the graph of f compressed horizontally toward the y axis by a factor of 1/c. If c < 1, the graph of g(x) = f( c· x ) is the graph of f stretched horizontally toward the y axis by a factor of 1/c.

16. Example 5 Q. Graph h(x) = [2x]. Answer: The parent function of h(x) is f(x) = [x]. The graph of the function h(x) compressed horizontally, toward the y – axis, by a factor of 1/2.

17. Word Problem 1 Q. Graph function g(x) = √x+4 and its parent function on the same set of axis. Answer: 1. Find the parent function of g(x): f(x) = √x 2. Graph the parent function 3. Graph g(x) by shifting 4 left from the parent function.

18. Word Problem 2 Q. Write a rule for the function whose graph can be obtained from the given parent function by performing the given transformations. Parent function: f(x) = x 3 Transformations: Shift the graph 5 units to the left and upward 4 units. A.1. Find a function that is shifted 5 units to the left: g(x) =(x+5) 3 2. Find a function that is shifted 5 units to the left and upward 4 units: g(x) = (x+5) 3 +4

19. Word Problem 3 Q. Describe a sequence of transformations that transform the graph of the parent function f into the graph of the function g. f(x) = x g(x) = -3(x-4)+1 A.1. Find how much did the f(x) shift in order to be g(x): Shift f(x) 4 units to the right and 1 units upward. 2. Find how much did the f(x) reflect in order to be g(x): Reflect f(x) across the x-axis. 3. Find how did the f(x) stretched: f(x) stretched vertically by a factor of 3 in order to be g(x).

20. Word Problem 4 Q. Graph each function and its parent function on the same graph. f(x) = -2(x - 1) 2 - 3 A.1. Find the parent function of f(x): f(x) = x 2 2. Find shifting: shift 1to right, and shift 3 downward. 3. Find the reflection: reflect vertically across x-axis. 4. Find the stretching and contraction: contract by a factor of 2.

21. Word Problem 5 Q. In 2002, the cost of sending first-class mail was $3.07 for a letter weighing less than 1 ounce, $0.60 for a letter weighting at least one ounce, but less than 2 ounces, $0.83 for a letter weighting at least 2 ounces, but less than 3 ounces, and so on. Write a function c(x) that gives the cost of mailing a letter weighting x ounces, and graph it. A. 1.Find the function: c(x) = 23[x] +37 2.Find parent function: f(x) = [x] 3.Find shifting, stretching, contraction, and reflection. 4.Shift 37 upward, and contract by a factor of 23.

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