Daily Warm Up Match each function with its graph.

Transformations of Graphs of Linear Functions Notes 3.6 Goals: Translate & Reflect graphs of linear functions Stretch & shrink graphs of linear functions Combine transformations of graphs of linear functions

Recall Core Vocabulary Linear Function Characteristics: an equation that can be written in the form f(x) = mx +b m= slope & b = y-intercept Most basic Linear Function: f(x) = x Straight Line Graph Quick review student’s don’t need to write down again. Use Popsicle Sticks to call on students to recall information. “if you did not remember this core vocabulary word, please write it down again”

Core Vocabulary Family of Functions— a group of functions with similar characteristics Parent Function— the most basic function in a family of functions the parent function of all linear functions is f(x) = x

Core Vocabulary Transformation— changes the size, shape, position, or orientation of a graph. Types of Transformations Translation Reflection Horizontal Shrink or Stretch Vertical Stretch or Shrink Skip a line

a transformation that shifts a graph horizontally or vertically Core Vocabulary Translation a transformation that shifts a graph horizontally or vertically Students should skip two lines from the definition of transformations.

Horizontal Translation Core Vocabulary Horizontal Translation Explain that the + comes from the double negative

Example 1.A. Let f(x) = x. Graph g(x) = f(x+2). Describe the transformations from graph of f to the graph of g. g(x) What is h? h = -2 What is the translation? Shift two units left. f(x)

Example 1.B. Let f(x) = x. Graph g(x) = f(x-5). Describe the transformations from graph of f to the graph of g. What is h? h = 5 What is the translation? Shift five units right. f(x) g(x)

Core Vocabulary Vertical Translation Explain that the + comes from the double negative

Example 1.C. Let f(x) = x. Graph g(x) = f(x) + 3. Describe the transformations from graph of f to the graph of g. g(x) What is k? k = 3 What is the translation? Shift three units up. f(x)

Example 1.D. Let f(x) = x. Graph g(x) = f(x) -6. Describe the transformations from graph of f to the graph of g. What is k? k = -6 What is the translation? Shift six units down. f(x) g(x)

You Try Let f(x) = 2x – 1. Graph g(x) = f(x) + 3. Describe the transformations from graph of f to the graph of g.

You Try Let f(x) = 2x – 1. Graph t(x) = f(x + 3). Describe the transformations from graph of f to the graph of t.

Core Vocabulary Reflection a transformation that flips a graph over a line called the line of reflection

Example 2.A. Let f(x) = x + 1. Graph g(x) = -f(x). Describe the transformations from graph of f to the graph of g. Reflect about the x-axis. Make a table of values to help students see the tranformation.

Example 2.B. Let f(x) = x + 1. Graph t(x) = f(-x). Describe the transformations from graph of f to the graph of t. Reflect about the y-axis.

Horizontal & Vertical Stretches & Shrinks Core Vocabulary Horizontal & Vertical Stretches & Shrinks

Example 3.A. Let f(x) = x – 1 . Graph g(x) = f( x). Describe the transformations from graph of f to the graph of g. X -3 3 1/3(x) 1/3(-3) = -1 1/3 (0) = 0 1/3 (3) =1 f(1/3 x) -1 – 1 = -2 0 – 1 = -1 1 – 1 = 0 Make a table of values to help students see the tranformation.

Example 3.B. Let f(x) = x – 1 . Graph t(x) = 3f(x). Describe the transformations from graph of f to the graph of t. X 1 2 f(x) 0-1 = -1 1-1 = 0 2 – 1 =1 3f(x) 3 * -1 = -3 3 * 0 = 0 3 * 1 = 3 Make a table of values to help students see the tranformation.

Example 4.A. Let f(x) = x + 2 . Graph g(x) = f(x). Describe the transformations from graph of f to the graph of g. Make a table of values to help students see the tranformation.

Example 4.B. Let f(x) = x + 2 . Graph t(x) = f(4x). Describe the transformations from graph of f to the graph of t. Make a table of values to help students see the tranformation.

Combining Transformations Core Concept Combining Transformations Step 1: Translate the graph horizontally h units . Step 2: Use “a” to stretch or shrink the resulting graph from step 1. . Step 3: Reflect the graph from step 2 when a<0 (when a is negative) . Step 4: Translate the graph from step 3 vertically k units

Example 5. Let f(x) = x. Graph h(x)= –2x + 3. Describe the transformations from graph of f to the graph of h. Make a table of values to help students see the tranformation.

Example 6. A cable company charges customers $60 per month for its service, with no installation fee. The cost to a customer is represented by c(m) = 60m, where m is the number of months of service. To attract new customers, the cable company reduces the monthly fee to $30 but adds an installation fee of $45. The cost to a new customer is represented by r(m) = 30m +45, where m is the number of months of service. Describe the transformations from the graph of c to the graph of r. Make a table of values to help students see the tranformation.

Example 6. Solution: Transformation from c(m) to r(m) We are going to GUESS. Givens: m is the # of months of service c(m) = 60m r(m) = 30m + 45 Unknowns: transformation from c(m) to r(m) Equations: Solve: graph c(m) & r(m) Solution: Transformation from c(m) to r(m) Translation up 45 units Vertical stretch of a factor of ½ Make a table of values to help students see the tranformation.