Name Period Exploring Graphs of Polynomial Functions Instructions: You will be responsible for completing this packet by the end of the period. You will have to read instructions for this activity. Please be sure you have actually read the instructions before you raise your hand to get help from your teacher. It s a-okay to ask your neighbor for clarification, too! After completing each page, DO raise your hand so you can get your TEACHER STAMP for the page. You must get each page stamped before moving on to the next page. Let s start this fun madness we call graphing! Go to the website www.desmos.com. Click on the red Launch Calculator button. After you finish, share your opinion of this activity. How could we make it better? What did you like? What did you hate? Did you learn stuff?
Activity 1: Desmos Basics Using Desmos, graph the function ( ) ( ). To graph this function, type the equation as shown into box number 1 on the left. To input the fraction, simply type 1/6 (then arrow over to end the fraction) and Desmos will convert it to. To put in exponents, you will use the ^ button (shift-6). TYPE: f(x)=1/6 (Now arrow over ) (x^5-10x^3+9x). It should look like this: As you are typing in the function, your graph should appear on the right. Desmos can generate a table to go with the graph. To get the corresponding table, click on the gear symbol above where you typed in the function: A couple of symbols will appear to the right of your function. The first symbol looks like a table, and it is! Click on the table, and Desmos will generate a table of values. Desmos automatically chooses 5 input values, but you can also input your own x-values into the table. Add a couple of x-values to your table, by clicking just below the 2 in the x-column of the table. Go ahead and add 3 and 5 to the table (below the 2 ). Desmos will automatically calculate the output (or y-value). Fill in the table below using the data from your Desmos table. ( ) Plot the points from your table on the coordinate plane to the right. If you feel like you need additional points to make your graph accurate, feel free to add more input values to your table. After plotting the points, draw the smooth curve in as accurately as you can. Once you have your graph complete, have your teacher stamp your page. After getting stamped, you are ready to move on to Activity 2. You can delete the function and table by clicking on the x on the right side of the function and table. Stamp #1
Activity 2: Adding a Constant to a Polynomial Function Let s get a fresh slate. Delete all equations and tables! Consider the function ( ). Graph the function in Desmos, sketch the graph, and answer the questions below. 1. Does this function have even or odd degree? Positive or negative leading coefficient? 2. How many zeros does f(x) have? Remember the Fundamental Theorem of Algebra 3. How many x-intercepts does the function have? 4. How can you explain the difference between your answer to #2 and your answer to #3? 5. List all the zeros of ( ), including multiplicities, if any. In Desmos, in box 2 & 3 (below ( )) type in the following equations: ( ) ( ) and ( ) ( ). There should now be three graphs on the coordinate plane (hopefully in three different colors). Answer the questions below. 6. Describe any differences (or change) between ( ) and ( ). 10. Describe any differences (or change) between ( ) and ( ). 7. How many total complex roots (or zeros) does the function ( ) have? Remember the Fundamental Theorem of Algebra 11. How many total complex roots (or zeros) does the function ( ) have? Remember the Fundamental Theorem of Algebra 8. List all the real zeros of ( ), including multiplicities, if any. How many non-real complex solutions does ( )have? Real Zeros: # Non-real solutions: 12. List all the real zeros of ( ), including multiplicities, if any. How many non-real complex solutions does ( )have? Real Zeros: # Non-real solutions: 9. Write ( ) in standard form. 13. Write ( ) in standard form. What does adding a constant to a polynomial function do to the graph? Do you remember the math-y word for this? Super bonus challenge question: Is there a constant we could add to ( ) so that we would know that none of the zeros of the functions were real numbers? Stamp #2
Activity 3: Adding a Constant to the Input (x) Fresh slate! Delete everything! Consider the function ( ). Graph the function in Desmos, sketch the graph, and answer the questions below. 14. Does this function have even or odd degree? Positive or negative leading coefficient? 15. How many total complex zeros does f(x) have? Remember the Fundamental Theorem of Algebra 16. How many real zeros (x-intercepts) does the function have? 17. List all the zeros of ( ), including multiplicities, if any. 18. Just for funsies, how many total complex zeros and how many real zeros does ( ) have? Complex zeros: Real Zeros: In Desmos, in box 2 & 3 (below ( )) type in the following equations: ( ) ( ) and ( ) ( ). There should now be three graphs on the coordinate plane (hopefully in three different colors). Answer the questions below. 19. Describe any differences between ( ) and ( ). (i.e. What did adding to do to the graph?) 22. Describe any differences between ( ) and ( ). (i.e. What did adding to do to the graph?) 20. How many total complex zeros does the function ( ) have? Remember the Fundamental Theorem of Algebra 23. How many total complex zeros does the function ( ) have? Remember the Fundamental Theorem of Algebra 21. What is the relationship between the zeros of ( ) and the zeros of ( )? 24. What is the relationship between the zeros of ( ) and the zeros of ( )? What does adding a constant to the input of a function do to the graph? Do you remember the math-y word for this? Super bonus challenge question: Can you write an equation for ( ) in terms of ( )? In other words how would you shift ( ) to get ( )? Stamp #3
Horizontal Translations Vertical Translations Activity 4: Thinking About the Stuff from Activity 2 & 3 (Plus What if we multiply a function by a constant instead of adding?) Use the graph of each function to help you answer each question. 25. Could we add a constant to the polynomial function ( ) such that it would have no real zeros? (Hint: real zeros are x- intercepts.) If yes, what is the smallest number you could add? 26. Could we add a constant to the function ( ) such that it would have no real zeros? If yes, what is could you add so that there were no real zeros? 27. What about this function? Is there a constant we can add so that there are no real zeros? 28. What about this function? Can I add a constant so that there are no real zeros? 29. Graph ( ) Without changing the graph, tell what the x-intercepts of ( ) will be. 30. Same function as 29. Without changing the graph, tell what the relative minimum of ( ) will be. 31. ( ) ( )( )( ) What will the zeros of ( ) be? What if we multiply a function by a constant instead of adding? 32. Graph ( ) and ( ). What do you notice? 33. Now graph ( ). [ this is the same as ( )] What do you notice? 34. If I wanted a function with the same x-intercepts as ( ), but a y-intercept of, what should I do to the function? 35. If I wanted a function with the same x-intercepts as ( ), but a y-intercept of, what should I do to the function? Super bonus challenge (for everyone!) 36. Function g(x) is shown. What are the zeros of ( )? 37. Function ( ) is shown. What is the y-intercept of ( )? Stamp #4
Activity 5: Factor by Graphing (What?!? You mean the factored form of a function and the graph of a function are somehow related?!? That s crazy talk! Not really, it s true.) 38. Graph the function ( ). (You may need to zoom out.) What are the x-intercepts? What are the zeros of ( ), including multiplicity, if any? What are the factors associated with each of those zeros? Write those factors all together. Wow! You just factored ( )! Amazing! 39. Graph the function ( ). Now can you factor it? See #38 above if you need help. 40. What about this: ( ). Can you at least find one real factor? What are the two nonreal complex solutions? ( Hint: use quad formula ) How are the graph of a function and the factored form of a function related? Super Bonus Challenge Question: Can you factor this insane function: ( ) (?!? Ummmm, please don t lose that GCF of. It should still be there in the factored form Stamp #5
Activity 6: So you think you are awesome Well you are. Now draw a picture. (Domain Restrictions) Click on question mark on the top right side of your screen. Then click on restrictions. This will give you a little tutorial on how to limit the domain of your function so that you just get a piece of it instead of the whole thing. Use domain restrictions and polynomial or other functions to create some art. You could try to write your name with functions, or make a face, or something abstract would be cool too. Whatever. The sky is the limit. Be sure to print your creation. You can save it if you like, by creating a Desmos log-in.