Problem Set 1: Transformations of Functions ... - SLIDEBLAST.COM
Problem Set 1: Transformations of Functions Directions: Complete the following problems on a separate sheet of graph or lined paper. You do not need to copy down the entire question, but make sure to write the number of each question. When you finish, mark all of your answers with a CHECK (correct) or CIRCLE (incorrect) and make any corrections needed on a separate, clean sheet. Key Terms:
Parent function Family of functions Horizontal translation (shift) Vertical translation (shift) Horizontal compression / stretch
Vertical compression / stretch Reflection across the y-axis (horizontal reflection) Reflection across the x-axis (vertical reflection)
I. Identify Transformations Given Equations of Functions 1. Using a table of values, sketch the graph of: 𝑓(𝑥) = 𝑥 2. You may use the inputs: 𝑥 = −2, −1, 0, 1, 2. What type of function is 𝑓(𝑥)? What is the shape of its graph called? DEFINITION: 𝑓(𝑥) in this example is called the PARENT FUNCTION. In the next question, we will create other functions in the same FAMILY OF FUNCTIONS.
2. Sketch a graph of each function below AND list the transformation(s) from the parent function, 𝑓(𝑥) = 𝑥 2 . Use a separate set of axes for each graph. a. b. c. d. e. f. g.
𝑔(𝑥) = 𝑥 2 + 5 𝑚(𝑥) = (𝑥 − 3)2 𝑛(𝑥) = (𝑥 + 2)2 − 4 𝑝(𝑥) = 2𝑥 2 𝑞(𝑥) = −3𝑥 2 + 4 𝑟(𝑥) = −(𝑥 + 1)2 − 2 1 𝑐(𝑥) = 5 − 2 (𝑥 + 2)2
HINTS: Create a table of values if stuck! You may need to use notes as well. Use Desmos to check your graphs. WRITE: Given the equation of a function, how can you tell the difference between a vertical shift and a horizontal shift? Be specific! Why is this true?
3. For each part, state all of the transformations taking place. Use precise language! Sketch a graph if needed. a. Describe in words the transformation(s) of the graph of 𝑓(𝑥) = (𝑥 − 3)2 + 1 to 𝑔(𝑥) = 𝑥 2 − 4 1 3
b. Describe in words the transformation(s) of the graph of 𝑓(𝑥) = (𝑥 + 3)2 to 𝑔(𝑥) = − (𝑥 + 3)2 c. Describe in words the transformation(s) of the graph of 𝑓(𝑥) = (𝑥 + 2)2 − 1 to 𝑔(𝑥) = −2(𝑥 − 3)2 + 5 (HINT: There are four different transformations!) d. Omar started with the graph of 𝑓(𝑥) = 𝑥 2 . He translated the graph up 3 units and to the left 4 units. What is the function equation 𝑔(𝑥) of his new graph? e. Angela started with the graph of 𝑓(𝑥) = 𝑥 2 . She translated the graph down 2 units, reflected the graph across the x-axis, and stretched the graph by a factor of 3. What is the function equation 𝑔(𝑥) of her new graph?
4. [CHALLENGE] Given a quadratic function in vertex form (see below), answer the following questions. DEFINITION: A quadratic function in VERTEX FORM is written as: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 a. b. c. d. e.
For what values of 𝑎 does a VERTICAL STRETCH take place? For what values of 𝑎 does a VERTICAL COMPRESSION take place? How does a VERTICAL STRETCH compare to a HORIZONTAL COMPRESSION? For what values of 𝑎 does a REFLECTION across the x-axis take place? Given a quadratic function in vertex form, what are the coordinates of its vertex? How can you find the coordinates of the vertex using transformations from the parent function?
II. Identify Transformations Given Graphs of Functions 5. For each part below, you are given a graph of the parent function and a transformed function of 𝑓(𝑥) = 𝑥 2. Write the function equation of the transformed graph, 𝑔(𝑥). Then list any transformations that have taken place from the parent function to the transformed function. a.
b.
6. Three quadratic functions are graphed below. Label each graph (widest, middle, narrowest) with its correct ′𝑎′ 1 2
value. Remember that ′𝑎′ represents the vertical stretch of a function. The options are: 𝑎 = 1, 𝑎 = , 𝑎 = 3.
WRITE: What would the graph of a quadratic function look like if its ′𝑎′ value was zero?
7. Below is a graph of the parent function and a transformed function of the function: 𝑓(𝑥) = |𝑥|. Write the function equation of the transformed graph, 𝑔(𝑥). Then list any transformations that have taken place from the parent function to the transformed function. HINT: 𝑓(𝑥) = |𝑥| is called the absolute value function. If you aren’t sure how we got its graph, then try it on your own using a table of values with the inputs: 𝑥 = −2, −1, 0, 1, 2.
III. Identify Transformations in General Form For all of the questions above, we have been focusing only on transforming quadratic functions of the form: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘. All of these functions have been transformations of the parent function: 𝑓(𝑥) = 𝑥 2 . However, we can transform any type of function, or even a general function 𝑓(𝑥) whose equation we don’t know. For example, we could transform 𝑓(𝑥) into 𝑔(𝑥) using the same general form, where: 𝑔(𝑥) = 𝑎𝑓(𝑥 − ℎ) + 𝑘 8. In the general form above, 𝑎, ℎ, 𝑎𝑛𝑑 𝑘 all affect the function 𝑓(𝑥) in different ways. How does: a. ′𝑎′ transform the function 𝑓(𝑥) (HINT: There are two different ways!)? b. ′ℎ′ transorm the function 𝑓(𝑥)? c. ′𝑘′ transform the function 𝑓(𝑥)?
9. Given a parent function 𝑓(𝑥) without its equation, list all of the transformations taking place for each part. a. b. c. d. e. f.
𝑔(𝑥) = 𝑓(𝑥) − 8 𝑔(𝑥) = 𝑓(𝑥 − 3) 𝑔(𝑥) = 𝑓(𝑥 + 2) − 4 𝑔(𝑥) = 3𝑓(𝑥) 𝑔(𝑥) = −𝑓(𝑥 − 1) + 1 1 𝑔(𝑥) = 10 − 4 𝑓(𝑥 + 8)
10. [CHALLENGE] Using the parent function 𝑓(𝑥) = 𝑥, graph the transformed function 𝑔(𝑥) in each case. You do not need to list the transformations. HINT: If you get stuck, just remember that 𝑓(𝑥) = 𝑥 means that you can substitute 𝑓(′𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔′) with ‘𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔′ any time you see 𝑓(′𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔′ ) in the equation. So re-write each function equation without 𝑓 first! e.g. 𝑔(𝑥) = 𝑓(𝑥 + 4) − 2 = (𝑥 + 4) − 2 = 𝑥 + 4 − 2 = 𝑥 + 2, so 𝑔(𝑥) = 𝑥 + 2 a. b. c. d. e. f.
𝑔(𝑥) = 𝑓(𝑥) − 5 𝑔(𝑥) = 𝑓(𝑥 − 5) 𝑔(𝑥) = 𝑓(𝑥 + 1) + 3 𝑔(𝑥) = −𝑓(𝑥) 𝑔(𝑥) = 2𝑓(𝑥) − 2 1 𝑔(𝑥) = 5 − 𝑓(𝑥 + 6) 3
WRITE: What type of function is 𝑓(𝑥) = 𝑥? What do you notice about the graphs in in part (a) and part (b)? Why did this happen?
*** Congratulations! You’ve completed your first Problem Set in Math II. Go find and high five your teacher! ***
Parent function Family of functions Horizontal translation (shift) Vertical translation (shift) Horizontal compression / stretch
Vertical compression / stretch Reflection across the y-axis (horizontal reflection) Reflection across the x-axis (vertical reflection)
I. Identify Transformations Given Equations of Functions 1. Using a table of values, sketch the graph of: 𝑓(𝑥) = 𝑥 2. You may use the inputs: 𝑥 = −2, −1, 0, 1, 2. What type of function is 𝑓(𝑥)? What is the shape of its graph called? DEFINITION: 𝑓(𝑥) in this example is called the PARENT FUNCTION. In the next question, we will create other functions in the same FAMILY OF FUNCTIONS.
2. Sketch a graph of each function below AND list the transformation(s) from the parent function, 𝑓(𝑥) = 𝑥 2 . Use a separate set of axes for each graph. a. b. c. d. e. f. g.
𝑔(𝑥) = 𝑥 2 + 5 𝑚(𝑥) = (𝑥 − 3)2 𝑛(𝑥) = (𝑥 + 2)2 − 4 𝑝(𝑥) = 2𝑥 2 𝑞(𝑥) = −3𝑥 2 + 4 𝑟(𝑥) = −(𝑥 + 1)2 − 2 1 𝑐(𝑥) = 5 − 2 (𝑥 + 2)2
HINTS: Create a table of values if stuck! You may need to use notes as well. Use Desmos to check your graphs. WRITE: Given the equation of a function, how can you tell the difference between a vertical shift and a horizontal shift? Be specific! Why is this true?
3. For each part, state all of the transformations taking place. Use precise language! Sketch a graph if needed. a. Describe in words the transformation(s) of the graph of 𝑓(𝑥) = (𝑥 − 3)2 + 1 to 𝑔(𝑥) = 𝑥 2 − 4 1 3
b. Describe in words the transformation(s) of the graph of 𝑓(𝑥) = (𝑥 + 3)2 to 𝑔(𝑥) = − (𝑥 + 3)2 c. Describe in words the transformation(s) of the graph of 𝑓(𝑥) = (𝑥 + 2)2 − 1 to 𝑔(𝑥) = −2(𝑥 − 3)2 + 5 (HINT: There are four different transformations!) d. Omar started with the graph of 𝑓(𝑥) = 𝑥 2 . He translated the graph up 3 units and to the left 4 units. What is the function equation 𝑔(𝑥) of his new graph? e. Angela started with the graph of 𝑓(𝑥) = 𝑥 2 . She translated the graph down 2 units, reflected the graph across the x-axis, and stretched the graph by a factor of 3. What is the function equation 𝑔(𝑥) of her new graph?
4. [CHALLENGE] Given a quadratic function in vertex form (see below), answer the following questions. DEFINITION: A quadratic function in VERTEX FORM is written as: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 a. b. c. d. e.
For what values of 𝑎 does a VERTICAL STRETCH take place? For what values of 𝑎 does a VERTICAL COMPRESSION take place? How does a VERTICAL STRETCH compare to a HORIZONTAL COMPRESSION? For what values of 𝑎 does a REFLECTION across the x-axis take place? Given a quadratic function in vertex form, what are the coordinates of its vertex? How can you find the coordinates of the vertex using transformations from the parent function?
II. Identify Transformations Given Graphs of Functions 5. For each part below, you are given a graph of the parent function and a transformed function of 𝑓(𝑥) = 𝑥 2. Write the function equation of the transformed graph, 𝑔(𝑥). Then list any transformations that have taken place from the parent function to the transformed function. a.
b.
6. Three quadratic functions are graphed below. Label each graph (widest, middle, narrowest) with its correct ′𝑎′ 1 2
value. Remember that ′𝑎′ represents the vertical stretch of a function. The options are: 𝑎 = 1, 𝑎 = , 𝑎 = 3.
WRITE: What would the graph of a quadratic function look like if its ′𝑎′ value was zero?
7. Below is a graph of the parent function and a transformed function of the function: 𝑓(𝑥) = |𝑥|. Write the function equation of the transformed graph, 𝑔(𝑥). Then list any transformations that have taken place from the parent function to the transformed function. HINT: 𝑓(𝑥) = |𝑥| is called the absolute value function. If you aren’t sure how we got its graph, then try it on your own using a table of values with the inputs: 𝑥 = −2, −1, 0, 1, 2.
III. Identify Transformations in General Form For all of the questions above, we have been focusing only on transforming quadratic functions of the form: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘. All of these functions have been transformations of the parent function: 𝑓(𝑥) = 𝑥 2 . However, we can transform any type of function, or even a general function 𝑓(𝑥) whose equation we don’t know. For example, we could transform 𝑓(𝑥) into 𝑔(𝑥) using the same general form, where: 𝑔(𝑥) = 𝑎𝑓(𝑥 − ℎ) + 𝑘 8. In the general form above, 𝑎, ℎ, 𝑎𝑛𝑑 𝑘 all affect the function 𝑓(𝑥) in different ways. How does: a. ′𝑎′ transform the function 𝑓(𝑥) (HINT: There are two different ways!)? b. ′ℎ′ transorm the function 𝑓(𝑥)? c. ′𝑘′ transform the function 𝑓(𝑥)?
9. Given a parent function 𝑓(𝑥) without its equation, list all of the transformations taking place for each part. a. b. c. d. e. f.
𝑔(𝑥) = 𝑓(𝑥) − 8 𝑔(𝑥) = 𝑓(𝑥 − 3) 𝑔(𝑥) = 𝑓(𝑥 + 2) − 4 𝑔(𝑥) = 3𝑓(𝑥) 𝑔(𝑥) = −𝑓(𝑥 − 1) + 1 1 𝑔(𝑥) = 10 − 4 𝑓(𝑥 + 8)
10. [CHALLENGE] Using the parent function 𝑓(𝑥) = 𝑥, graph the transformed function 𝑔(𝑥) in each case. You do not need to list the transformations. HINT: If you get stuck, just remember that 𝑓(𝑥) = 𝑥 means that you can substitute 𝑓(′𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔′) with ‘𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔′ any time you see 𝑓(′𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔′ ) in the equation. So re-write each function equation without 𝑓 first! e.g. 𝑔(𝑥) = 𝑓(𝑥 + 4) − 2 = (𝑥 + 4) − 2 = 𝑥 + 4 − 2 = 𝑥 + 2, so 𝑔(𝑥) = 𝑥 + 2 a. b. c. d. e. f.
𝑔(𝑥) = 𝑓(𝑥) − 5 𝑔(𝑥) = 𝑓(𝑥 − 5) 𝑔(𝑥) = 𝑓(𝑥 + 1) + 3 𝑔(𝑥) = −𝑓(𝑥) 𝑔(𝑥) = 2𝑓(𝑥) − 2 1 𝑔(𝑥) = 5 − 𝑓(𝑥 + 6) 3
WRITE: What type of function is 𝑓(𝑥) = 𝑥? What do you notice about the graphs in in part (a) and part (b)? Why did this happen?
*** Congratulations! You’ve completed your first Problem Set in Math II. Go find and high five your teacher! ***
