Lesson 7 - Using Transformations to Graph Functions.pdf

Throughout the unit, we have reviewed the basic transformations of functions, and have also learned some new ones. At this point, we should be familiar with the following transformations of functions: Vertical Stretch Vertical Compression Reflection in the x-axis Reflecti...

Throughout the unit, we have reviewed the basic transformations of functions, and have also learned some new ones. At this point, we should be familiar with the following transformations of functions: Vertical Stretch Vertical Compression Reflection in the x-axis Reflection in the y-axis Horizontal Stretch Horizontal Compression Translation In this lesson, we will be putting all of these transformations together to graph functions. Function Transformations LEARNING QR CODES GOALS Given any parent function y = f ( x), transformations of the function are Number Sets Determine, through investigation using technology, given by: the roles of the parameters a, k, d, and c in functions and y = af [k ( x − d )] + c describe these roles in terms of transformations on the graphs Sketch graphs by applying one or more transformations to the graphs of parent functions, then state the domain and Interval vs Set Notation range of the transformed function Graph piecewise functions comprised of a variety of different familiar functions Describe how different transformations applied to a parent function affect either the domain or range of the parent function.  Example 1 – Graphing Parabolas Sketch the graph of f ( x) = x 2 and 1 the graph of f ( x) = ( x + 4) 2 − 2 on 2 the same grid. Example 2 – Graphing the Square Root Function Given f ( x) = x sketch the graph of y = f ( x) and the graph of y = 2 f (− x − 3) + 4 on the same grid  Example 3 – Graphing the Absolute Value Function Given f ( x) = | x | sketch the graph of 1 y = − f (−0.25 x − 1) − 2 2 Example 4 – Graphing the Reciprocal Function 1 Given f ( x) = sketch the graph of x y = f ( x) and the graph of y = − f (2 x − 4) − 1 on the same grid Piecewise Functions A piecewise function is a special type of function that is made up of many individual pieces. Last year, you would have graphed some piecewise functions using linear and quadratic components, but now we have many more graphs to work with! Piecewise Functions can be graphed using the following strategy: 1. Identify how many “pieces” you will have to x +1 − 5  x  −3 graph, and identify the type(s) of parent  f ( x) = −( x + 1) 2 + 2 − 3  x  0 functions that each piece will be.   x+2 x0 2. Starting with the first piece, graph the function as normal. For instance, if the first piece is a parabola, draw a graph of the parabola. It may be a good idea to do it very lightly in pencil. 3. Once graphed, look at the domain that is specified in the piecewise function. Erase the graph you’ve drawn anywhere where that piece shouldn’t exist. 4. Repeat the above steps for each piece of the function. 5. Note that if the endpoint(s) of a function do not land on a “base point” for that graph, you should calculate the value at that endpoint. This will make it easier for you to graph.  Example 5 – Graphing Piecewise Functions Sketch the graph of the following piecewise function. x +1 − 5  x  −3  f ( x) = −( x + 1) + 2 − 3  x  0 2   x+2 x0 Example 6 – Graphing Piecewise Functions Sketch the graph of the following piecewise function.  2( x + 3) 2 − 4 x  −2  f ( x) =  x − 1 −2 x  2 1  ( x − 3) x2  3  Summary of Parent Functions Parent Function General Transformed How transformations affect How transformations affect Equation, Sketch and Domain Range Form Domain Range Base Points Linear: f ( x) = x Range is not affected. {x  } f ( x) = a[k ( x − d )] + c {y  } Quadratic: f ( x) = x 2 {y  | y  0} Absolute Value: f ( x) = x 2 Domain is not affected. {y  | y  0} {x  } Square Root: f ( x) = x Domain is affected by the sign of k and the value of d. {x  | x  0} If k is positive: {x  | x  d} If k is negative: {x  | x  d } 1 Reciprocal: f ( x) = x {x  | x  0}