Just found this blogpost in my drafts, way back from January! Now that it is summer I have time to finish it up. To play with this applet, move points A, B, and C so that they are on a transformation of the parent graph of y=x^2. The idea here is that students will be given a verbal or written description of a transformation, and then physically move points A, B, and C to the appropriate locations. Once you've moved the floating points to the correct location, guess the new function for your transformed parabola. Type into the input box to check your answer. If you are correct, the parabola will go through the 3 points and it will also turn green (plus you get a message saying you are correct). You can try the applet below, or here at Geogebratube.
I explored this in a previous blogpost about graphing the transformations of square and cube root graphs. What I like about this approach is that it requires an understanding of a function as a set of points. When I transform a function, I'm not just moving a shape around the grid. I'm actually transforming a set of points, applying the same transformation rule to all points on the original graph.
I'm still working through some design decisions, so feel free to comment below. First, should the floating points (A, B, and C) start on the parent graph? Second, will extra points be more helpful in exploring this concept? I made another copy of this worksheet with 5 floating points. This worksheet is available on Geogebratube here.
it is possible for students to get the wrong "answer" using this Geogebra Worksheet. As long as the function entered goes through the 5 points, the Worksheet will indicate that you are correct. So any type of function can be used. This has happened a few times since I started using applets like this one, and it's made for great conversation and reinforces the need to check the reasonableness of an answer.
Link to Geogebra Book with all 4 applets Last year was the first year we taught trig graphing in Algebra II, and so it was a brand new teaching topic for a couple of our teachers. Perfect opportunity for collaboration! We decided to introduce the topic using Dan Meyer's Ferris Wheel. This introduction built on intuition and gave students a day to practice reasoning with a context that involved sine and cosine graphs. The equation for a sine graph was introduced at the end of the lesson, and students used sliders in Geogebra to fit the sine graph to the ferris wheel data of time versus height. Towards the end of this unit students were asked to model real world situations using trigonometric functions. We picked a few problems involving tides and ended with a problem from Illustrative Mathematics called Foxes and Rabbits 2. We used a series of scaffolded Geogebra presentations to help students get started. The first presentation is below, and is an embedded applet that you can play with. Type a function into the f(x) input bar and press enter to see it on the graph. You should try this right now. It is fun! The goal is to get the sine graph to pass through all of the data points.
We introduced this to students by first talking about the data and how tides are measured. You can talk about how the moon impacts tides and how the data is roughly periodic. At this point students should know the amplitude and period for y=sin(x). Next, have a class conversation about how to transform this graph to pass through the given points. The power of a premade interactive model is that you can pause during the conversation to allow students time to process. I make use of the think-pair-share structure with the expectation that students may be called on to explain what they know or discussed with a partner. One of the teachers on the Algebra II team started writing and sharing the questions that she used with these presentations, and the other teachers found this to be very helpful. Below is a list of questions that can be used with this presentation after the initial conversation about the data. Feedback on this list of questions would be great! 1. What is the midline for the Santa Cruz Tides data? (Discuss first, then reveal the midline by selecting the midline box.)
2. How can I change (transform) the function f(x)=sin(x) so that it has this midline? (Discuss, then type in the correct function and reveal the change in the graph.)
3. What is the maximum value of this function? What does it represent with respect to the tides? Find the minimum value as well. Discuss and then reveal the max and min lines by selecting the appropriate box.
4. What is the difference between the highest and lowest tide measurements? Does this help me find the amplitude, period, or vertical shift of my function?
5. Now that I know that the amplitude is 1.5, how can I change my function f(x)=sin(x)+2.5 to account for an amplitude that is not 1?
6. Discuss with your partner what one period of y=sin(x) looks like. My current graph has been shifted horizontally from the parent graph y=sin(x). Can you find a new starting point? How can we change our function y=1.5sin(x)+2.5 so that it has been shifted horizontally to your new starting point? (There are multiple answers here, which can be discussed now or later depending time.)
7. What about the period? Note: Most classes will have a formula to use to account for a change in period. I haven't taught trig in years, so I just use horizontal stretch/shrink reasoning. y=sin(x) has a period of 2pi, and Santa Cruz tides has a period of 12. Since 12 is larger than 2pi, I will multiply x by a factor of (2pi)/12. My new function is f(x)=1.5sin(2pi/12(x-9))+2.5.
One of my favorite things about Common Core is the opportunity to learn new ways to introduce content. I have been teaching high school for 7 years, so I haven't had the chance to teach with too many books. With common core and the impending adoption of new materials, I am finding new ways to teach almost weekly! I had a chance this week to play around with domain and range using the TI Inspire. I can't take credit for most of the ideas below, but this introduction is too good not to share! Check out the Texas Instruments website for free worksheet downloads that go with domain and range presented in this manner, plus many other topics. The worksheets go with TI calculator activities, but you can still get great ideas from them if you don't have access to the technology. Drag the gray point in the presentation below over points A, B, C, D and E. Watch as the domain and range values appear on the axes. To try a different problem set, hit "clear domain and range", drag each point to a new location, and repeat the activity. Can you arrange the points so that the domain is {-2,3,5} and the range is {1,2}? This question has multiple solutions and will give students the opportunity to discuss the mathematics and check answers in real time. They are also practicing with multiple representations. Students can find domain and range of a line segment below by dragging point C. Again the domain and range values appear along the axes. To try a new problem hit "clear trace" and drag the endpoints to a new location. Can you find the line segment that has domain greater than or equal to -3 and less than 4, and range greater than -1 and less than or equal to 3? Insert inequality statements into these problems to help students practice with the multiple representations used in this context. For the presentation below drag point A to create a relation. Can you graph the relationship with domain and range described below? Is your relation a function?
To show domain and range for a specific function, enter the function into the input box below. Change the left and right limit of the function to make it piecewise, or to see what the domain and range are for the the function itself change the limits to values that are not on the screen below. (So -6 and 8 will work). Links you may find useful: Geogebra Tools from Making Math Visual blog. Geogebra Series II: Transforming a Graph With Sliders
Inverse functions is one of many concepts that can be conceptually enhanced through the dynamic features of Geogebra. Watch the short clip below to see some of the visual aspects that can be incorporated into your presentation.
Inverse Functions Tool I like this treatment of inverse functions because it starts off with points (solutions) for a given function and we begin by switching the x and y coordinates to find points for the inverse relation. As the value of "a" changes, students can see the different points of the relation being traced across the screen. For the first example I picked y=x^2. I graph the solutions for this relation by letting point A=(a, a^2). Slider a is set up to go from -10 to 10, so I can graph all of the points for the function y=x^2 for x between -10 and 10. To plot the points for the inverse relation, I simply switch the x and y-coordinates for point A. My second point is B=(a^2, a).
Once the two relations are graphed we can observe the reflection over the line y=x. We can predict points that will be on the graph of y=x^2, and then predict which points will be on the inverse relation. To do this, simply enter a point into the input bar at the bottom of the screen, and hit enter.
We can discuss what equations will create each graph. Type into the input bars for Point A and B functions, hit enter, and check the box in order to display each graph. This is a natural place in the lesson to introduce the procedure for finding inverses (switch x and y and solve for y), because students can immediately check answers when finished by graphing the inverse function.
When students notice that only half of the inverse relation is represented by y=sqrt(x), we talk about why the inverse relation is not a function. The answer isn't "because it fails the vertical line test", or "because the original function fails the horizontal line test". The answer is simply that the inverse relation isn't A function because it is TWO functions. We then add the second function of y=-sqrt(x). Type into the input bar at the bottom of the screen and hit enter to view this graph. When we use the vertical and horizontal line tests later as a tool to predict whether one or more functions exist for the inverse relation of a given function, it will be with a greater understanding of why these rules are true.
To try other examples using the Inverse Functions Tool refresh the screen. Some other examples are:
Point A=(a, a^3), Point B=(a^3,a)
Point A=(a, (a-2)^2-3), Point B=((a-2)^2-3,a)
Point A=(a, 2^a), Point B=(2^a,a)
Point A=(a, sin(a)), Point B=(sin(a),a)
For these examples you might plot the points for A first, make predictions about what the second relation will look like, whether it will be a function, and what the equations will be. We can then verify these predictions using Geogebra. For functions such as y=2^x that haven't been studied yet, we can reinforce the understanding of both solution set for an equation as well as inverse relation by choosing points to plot on each graph.
We can come back to this representation of inverse functions in math analysis or precalculus when more function types have been studied. We can also use this representation to teach specific types of inverse functions such as logarithms or inverse trigonometric functions.
I checked out a section of my Algebra II text on inverse functions, and I found 7 examples. 1. Find an equation of an inverse function. 2. Verify that two functions are inverses of each other. 3. A multi-step word problem 4. Find an inverse and graph both functions. Restrict the domain. 5. Graph a function, use horizontal line test to decide if inverse exists, then find inverse. 6. Find inverse of a power model (so y=a*x^b) 7. Use the previous inverse power model to make a prediction. This treatment of inverse functions relies heavily on algebraic manipulation and substitution. This section is also heavy on vocabulary and mathematical notation. Having a strong conceptual background grounded in a visual understanding of material can be helpful for these sections. Being able to access multiple representations of the material by using a tool like Geogebra can also be helpful. Links you may find useful: Geogebra Toolsfrom MakingMath Visual blog. Geogebra Wiki. Includes tutorials on how to get started with Geogebra. Some Things I Wish I Knew When I Started Using Geogebra.
One of the most powerful features of Geogebra is the slider. Watch the video below to see how sliders can make your graphs dynamic.
You can use the above tool for presentation in your classes, or you can open a new Geogebra file and follow the steps below to create your own presentation. The first step is to add a slider to the graphics view. Click in the top left or right of the graphics view to keep the slider out of the way of the graph that you will add later.
Use -5 to 5 or -10 to 10 for the minimum and maximum values. Now add sliders b, c, and d.
Choose your favorite function type, such as quadratics. I can graph as many quadratic functions as I want by typing each function into the input bar at the bottom of the screen. Or I can type one quadratic function into the input bar that can be transformed into any quadratic function (okay not all of them, just the ones with parameters between -5 and 5, or whatever you chose for your min and max values).
Use parameter b for horizontal stretch shrink.
Or for growth and decay
Hit enter, and prepare to be amazed! Drag each of your sliders to see how the function is transformed.
You can use this parent graph tool, which includes many types of functions. Once students see functions being transformed dynamically, the conversations and types of questions we can ask get richer.
If students know the form of the function (y=a(x-c)^2+d in this case), they can discuss the similarities and differences between the parabolas in the family above. They can notice that all of the parabolas have a vertex at (-2,-1), and an axis of symmetry at x=-2, and that there is a relationship between these values and the parameters c and d. This is a good opportunity to practice vocabulary, and also to help students start understanding the difference between a parameter and a variable. They can even use the parent graph tool to create their own graph family.
y=a*sin(b(x-c))+d
For the graphs above students might notice that the amplitudes are different and that a is between -5 and 0. For the graphs below they can recognize exponential growth (b>1), horizontal shifting (c varies), and maybe guess that the horizontal asymptote is y=0 (so d=0).
y=a*b^(x-c)+d
Once students have explored function transformations we can introduce other types of questions. One example is to graph a parent function and transformed function and ask students to make generalizations on the parameters. One of my favorite tricks is to take the numbers off of the axes so I can also ask them to write a possible function for the transformed graph.
If students know the form of the parent graph is y=a*abs(x-c)+d, they can reason that a<0, absolute value of a is less than 1, c<0, and d<0. And of course there is the classic matching activity.
This can be done with any type of function, but for this post I've chosen a cubic function. The following activity is intended for students to do on the computer after modeling a few possibilities with the whole class. http://www.geogebratube.org/student/m86165 Students are given a cubic function f, and asked to find four transformations of f that go through a given point. This is a great activity to help students practice using function notation to describe a transformation, and the use of technology provides them with immediate feedback.
