Parent Functions and Transformations Section 1.1 beginning on page 3

The Big Ideas In this section we will learn about…. Families of functions A family of functions is a group of functions that share the same key characteristics. The parent function is the most basic function in the family. Transformations Functions in the same family are transformations of the parent function. Changes to the parent function create specific changes to the graph of the function, these changes are consistent through the different families of functions.

Core Vocabulary Previously Learned: Function Domain Range Slope Scatter Plot New: Parent Function Transformation Translation Reflection Vertical Stretch Vertical Shrink

Basic Parent Functions Constant: Absolute Value: Parent Function: 𝑓 𝑥 =1 Domain: All Real Numbers Range: 𝑦=1 Parent Function: 𝑓 𝑥 = 𝑥 Domain: All Real Numbers Range: 𝑦≥0 Linear: Quadratic: Parent Function: 𝑓 𝑥 =𝑥 Domain: All Real Numbers Range: Parent Function: 𝑓 𝑥 = 𝑥 2 Domain: All Real Numbers Range: 𝑦≥0

Identifying a Function Family Example 1: Identify the function family to which 𝑓 belongs. Compare the graph of 𝑓 to the graph of its parent function. The graph of 𝑓 is V shaped, so 𝑓 is an absolute value function. The graph of 𝑓 is shifted up and is narrower than the graph of the parent function. The domain of 𝑓 is all real numbers. (same as the parent function) 𝑔 𝑥 = 𝑥 𝑓 𝑥 =2 𝑥 +2 The range of 𝒇 is 𝒚≥𝟏. (the range of the parent function is 𝑦≥0)

Identifying a Function Family Example 1: Identify the function family to which 𝑓 belongs. Compare the graph of 𝑓 to the graph of its parent function. The graph of 𝑓 is U shaped, so 𝑓 is a quadratic function. The graph of 𝑓 is shifted right and is widerthan the graph of the parent function. The domain of 𝑓 is all real numbers. (same as the parent function) 𝑓 𝑥 = 1 4 (𝑥−3) 2 𝑔 𝑥 = 𝑥 2 The range of 𝒇 is 𝒚≥𝟎. (same as the parent function)

Describing Transformations A transformation changes the size, shape, position, or orientation of a graph. A translation is a transformation that shifts a graph horizontally and/or vertically but does not chance its size, shape, or orientation. A reflection is a transformation that flips a graph over a line called the line of reflection. A reflected point is the same distance from the line of reflection as the original point but on the opposite side. Another way to transform a function is to multiply all of the y-coordinates by the same positive factor (other than 1). When the factor is greater than 1, the transformation is a vertical stretch. When the factor is greater than 0 and less than 1, it is a vertical shrink (also known as a vertical compression). *** we will learn about horizontal stretches and shrinks in the next section.

Describing Transformations Example 2: Graph 𝑔 𝑥 =𝑥−4 and its parent function. Then describe the transformation. The graph of 𝑔 is a linear function. 𝑓 𝑥 =𝑥 The graph of 𝑔 is 4 units below the graph of the parent linear function 𝑓. The graph of 𝑔 𝑥 =𝑥−4 is a vertical translation 4 units down of the graph of the parent linear function.

Describing Transformations Example 3: Graph 𝑔 𝑥 =− 𝑥 2 and its parent function. Then describe the transformation. The graph of 𝑔 is a quadratic function. Use a table of values to graph each function: 𝑓 𝑥 = 𝑥 2 𝒙 𝒚= 𝒙 𝟐 𝒚=− 𝒙 𝟐 -2 -1 1 2 4 −4 1 −1 1 −1 4 −4 The graph of g is the graph of the parent function flipped over the x-axis The graph of 𝑔 𝑥 =− 𝑥 2 is a reflection in the x-axis of the graph of the parent quadratic function.

Graphing and Describing Stretches and Shrinks Example 4 a : Graph 𝑔 𝑥 =2 𝑥 and its parent function. Then describe the transformation. The graph of 𝑔 is an absolute value function. Use a table of values to graph each function: 𝑓 𝑥 = 𝑥 𝒙 𝒚= 𝒙 𝒚=𝟐 𝒙 -2 -1 1 2 2 4 1 2 1 2 2 4 The y-coordinate of each point on g is two times the y-coordinate of the corresponding point on the parent function. The graph of 𝑔 𝑥 =2 𝑥 is a vertical stretch of the graph of the parent absolute value function.

Graphing and Describing Stretches and Shrinks Example 4 b : Graph 𝑔 𝑥 = 1 2 𝑥 2 and its parent function. Then describe the transformation. The graph of 𝑔 is a quadratic function. Use a table of values to graph each function: 𝑓 𝑥 = 𝑥 2 𝒙 𝒚= 𝒙 𝟐 𝒚= 𝟏 𝟐 𝒙 𝟐 -2 -1 1 2 4 2 1 0.5 1 0.5 4 2 The y-coordinate of each point on g is one-half of the y-coordinate of the corresponding point on the parent function. The graph of 𝑔 𝑥 = 1 2 𝑥 2 is a vertical shrink of the graph of the parent quadratic function.

Combinations of Transformations Example 5: Use a graphing calculator to graph 𝑔 𝑥 =− 𝑥+5 −3 and its parent function. Then describe the transformations. The function g is an absolute value function. The graph shows that 𝑔 𝑥 =− 𝑥+5 −3 is a reflection in the x-axis … … followed by a translation 5 units left and 3 units down of the graph of the parent absolute value function.

Combinations of Transformations Use a graphing calculator to graph the function and its parent function. Then describe the transformation. 8) ℎ 𝑥 = 1 4 𝑥+5 The function h is a linear function. The graph shows that h is a reflection in the x-axis followed by a vertical translation 5 units up and a vertical shrink.

Combinations of Transformations Use a graphing calculator to graph the function and its parent function. Then describe the transformation. 9) 𝑑 𝑥 =3 (𝑥−5) 2 −1 The function d is a quadratic function. The graph shows that h is translation 5 units right and 1 unit down, and a vertical stretch.

Modeling With Mathematics Example 6: The table shows the height y of a dirt bike x seconds after jumping off a ramp. What type of function can you use to model the data? Estimate the height after 1.75 seconds. You can model this data with a quadratic function. Time (seconds), x Height (feet), y 8 0.5 20 1 24 1.5 2 𝑥=1.75 (1.75, 15) The graph shows that the height is about 15 feet after 1.75 seconds.

Modeling With Mathematics The table shows the amount of fuel in a chainsaw over time. What time of function can you sue to model the data? When with the tank be empty? Time (minutes), x 10 20 30 40 Fuel remaining (fluid ounces), y 15 12 9 6 3 The tank will be empty when y=0. Use the graph to predict the value of x. You may have also detected a pattern in the table of values to help you predict the value of x. The tank will be empty after 50 minutes. You can model the data with a linear function.