Transformations of functions
- 1. Transformations of Functions Mr. Johnson Algebra 1 California Content Standard – Building Functions
- 2. Objectives & CA Content Standard • Students will learn the properties of rigid and non-rigid transformations on different types of parent functions and will be able to distinguish them via mathematical expression and graphical representation. CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
- 3. Before We Begin, Let’s Recap What do we know about functions? Spend 2-3 minutes discussing with your classmates in groups of 4 EVERYTHING you can think about regarding functions. We will then write down one thing from each group together as a class.
- 4. Using an xy Table to graph functions Let’s consider the function: 𝑓 𝑥 = 𝑥2 TABLE x y -2 4 -1 1 0 0 1 1 2 4 GRAPH
- 5. We can use these xy tables to get the graphs of what we call “Parent Functions” x y -2 -2 -1 -1 0 0 1 1 2 2 x y -2 4 -1 2 0 0 1 2 2 4 x y -2 2 -1 1 0 0 1 1 2 2 x y -2 -8 -1 1 0 0 1 1 2 8 x y 0 0 1 1 4 2 9 3 16 4 x y -2 2 -1 2 0 2 1 2 2 2 Linear Function 𝑓 𝑥 = 𝑥 Quadratic Function 𝑓 𝑥 = 𝑥2 Absolute Value Function 𝑓 𝑥 = |𝑥| Cubic Function 𝑓 𝑥 = 𝑥3 Square Root Function 𝑓 𝑥 = 𝑥 Constant Function 𝑓 𝑥 = 2
- 6. We can use these xy tables to get the graphs of what we call “Parent Functions” Linear Function 𝑓 𝑥 = 𝑥 Quadratic Function 𝑓 𝑥 = 𝑥2 Absolute Value Function 𝑓 𝑥 = |𝑥| Cubic Function 𝑓 𝑥 = 𝑥3 Square Root Function 𝑓 𝑥 = 𝑥 Constant Function 𝑓 𝑥 = 2
- 7. What happens if we add a one to the function? 𝑓 𝑥 = 𝑥2 + 1 What if we add 5? 𝑓 𝑥 = 𝑥2 + 5 How about if we subtract 2? 𝑓 𝑥 = 𝑥2 − 2 Revisiting the function 𝑓 𝑥 = 𝑥2
- 8. GROUP ACTIVITY! • Fill the following xy tables with your group mates and make a corresponding graph. Discuss with each other any similarities you have found between these three new quadratic functions and the parent function. x y -2 -1 0 1 2 x y -2 -1 0 1 2 x y -2 -1 0 1 2 𝑓 𝑥 = 𝑥2 + 1 𝑓 𝑥 = 𝑥2 + 5 𝑓 𝑥 = 𝑥2 − 2
- 9. What did you notice? • This is an example of a RIGID TRANSFORMATION • A RIGID TRANSFORMATION is a transformation in which the basic shape of the graph is unchanged • RIGID TRANSFORMATIONS change only the position of the graph in the coordinate plane
- 10. Types of RIGID TRANSFORMATIONS • There are 3 types of RIGID TRANSFORMATIONS 1. VERTICAL SHIFTS 2. HORIZONTAL SHIFTS 3. REFLECTIONS
- 11. VERTICAL & HORIZONTAL SHIFTS Let 𝑐 be some positive real number. We can determine VERTICAL & HORIZONTAL SHIFTS with the following formulas: VERTICAL SHIFT 𝑐 units upward: 𝑔 𝑥 = 𝑓 𝑥 + 𝑐 VERTICAL SHIFT 𝑐 units downward: 𝑔 𝑥 = 𝑓 𝑥 − 𝑐 HORIZONTAL SHIFT 𝑐 units to the right: 𝑔 𝑥 = 𝑓(𝑥 − 𝑐) HORIZONTAL SHIFT 𝑐 units to the left: 𝑔 𝑥 = 𝑓(𝑥 + 𝑐)
- 12. Example of HORIZONTAL SHIFT We explored some examples of VERTICAL SHIFTS in our groups. Let’s take a look at the function 𝑓 𝑥 = (𝑥 + 2)2 together! x y -4 4 -3 1 -2 0 1 1 0 4
- 13. REFLECTIONS A reflection will reflect your function, like a mirror, over the x or y axis. We can determine what type of reflection with the following formulas: Reflection over the x-axis: 𝑔 𝑥 = −𝑓(𝑥) Reflection over the y-axis: 𝑔 𝑥 = 𝑓(−𝑥)
- 14. Examples of REFLECTIONS Here are two ways the Square Root Function is being transformed via a REFLECTION. Do you notice anything wrong with these graphs? Discuss with your group why one of these reflections is not possible.
- 15. Now let’s look at what happens to the Quadratic Function if we multiply or divide by some constant 𝑐. What happens if we multiply by 4? 𝑓 𝑥 = 4𝑥2 What happens if we divide by 2? 𝑓 𝑥 = 𝑥2 2 Revisiting the function 𝑓 𝑥 = 𝑥2
- 16. Group Activity! Once again, fill out the xy tables below, then graph the two functions with your group mates. Discuss what different transformations occur when you multiply versus divide by a constant. x y -2 -1 0 1 2 x y -2 -1 0 1 2 𝑓 𝑥 = 4𝑥2 𝑓 𝑥 = 𝑥2 2
- 17. What did you notice? These types of transformations are called NON-RIGID TRANSFORMATIONS • A NON-RIGID TRANSFORMATION is a transformation of a graph that causes a change in the shape of the graph.
- 18. Types of NON-RIGID TRANSFORMATIONS • There are 4 types of NON-RIGID TRANSFORMATIONS 1. VERTICAL STRETCH 2. VERTICAL SHRINK 3. HORIZONTAL STRETCH 4. HORIZONTAL SHRINK
- 19. Stretches and Shrinks Let 𝑐 be some real number. We can determine VERTICAL & HORIZONTAL STRETCHES & SHRINKS with the following formulas: VERTICAL STRETCH: 𝑔 𝑥 = 𝑐𝑓 𝑥 , where 𝑐 > 1 VERTICAL SHRINK: 𝑔 𝑥 = 𝑐𝑓 𝑥 , where 0 < 𝑐 < 1 HORIZONTAL STRETCH: 𝑔 𝑥 = 𝑓(𝑐𝑥), where 𝑐 > 1 HORIZONTAL SHRINK: 𝑔 𝑥 = 𝑓(𝑐𝑥), where 0 < 𝑐 < 1
- 20. A Useful Tip! Stretches and Shrinks can be easily confused with one another. Here is a useful trick to help you remember which is which. • VERTICAL STRETCH: y-values increased to corresponding x-values • VERTICAL SHRINK: y-values decreased to corresponding x-values • HORIZONTAL STRETCH: x-values increased to corresponding y-values • HORIZONTAL SHRINK: x-values decreased to corresponding y-values
- 21. Example of HORIZONTAL SHIFT In your groups you looked at an example of a VERTICAL STRETCH and a VERTICAL SHRINK. Let’s look at an example of a HORIZONTAL STRETCH and HORIZONTAL SHRINK TOGETHER MATCH THESE TRANSFORMATIONS TO THE GRAPH 𝑓 𝑥 = 𝑥 𝑓 𝑥 = 4𝑥 𝑓 𝑥 = 1 4 𝑥
- 22. Group Activity! Putting It All Together It’s possible for a function to undertake more than one transformation at once. With your group mates, try graphing the following functions: 𝑓 𝑥 = −2𝑥2+3 𝑓 𝑥 = 1 2 𝑥 + 3 − 2 𝑓 𝑥 = 𝑥 − 2 + 1
- 23. Group Activity! Building A Function We now have the tools to interpret graphs and build the functions that create them. Take a look at the two graphs below. Try to come up with the function that maps them.
- 24. Summary • We just learned how introducing constants in various ways can transform a variety of different functions. As you continue your math career, you will learn about many more parent functions and these types of transformations will hold for all of them. This is a key concept that will be important as we continue to analyze different function topics.