Exploring Graphs of Power Functions Using the TI-Nspire

Size: px

Start display at page:

Download "Exploring Graphs of Power Functions Using the TI-Nspire"

Cassandra Richards
6 years ago
Views:

1 Exploring Graphs of Power Functions Using the TI-Nspire I. Exploration Write Up: Title: Investigating Graphs of Parabolas and Power Functions Statement of Mathematical Exploration: In this exploration, I will use the TI-Nspire CAS software to discover the effect of parameter changes on the graph of a power relation. In part 1, I will make sliders for a, b, and c for the equation y = a x! + c. I will pay attention to the movements and make conjectures about how each parameter affects the graph. Next I will explore three cases, in case one, b, c will be constant while a varies. Then in case two, a, c are constant and b varies. Finally in case three, a, b will be constant while c varies. There will be 6 functions on each graph to show what happens when the parameters vary. In part 2, I will explore the roots of 15 graphs that are on the same axis with the property that they have an x- intercept in common. Finally in part 3, I will create a 3D plot of a graph of a power function. I will create sliders for a, b, and c and investigate the change of the graph when they are changed. My Exploration: Part 1: 1. Slider Exploration Here I will graph the function y = a x! + c and then make sliders for a, b and c. Then I will pay attention to the movement of the function when I move each slider. When I moved slider a while b = 1, the function s slope changed. When a > 0, the function will have a positive slope. When the positive a increases the slope of the function increases in steepness. When a < 0, the function has a negative slope and as the negative slope decreases, the slope increases in steepness.

2 When I moved slider a while b = 2, the parabola s width changed. It shrunk when a was increasing and then it stretched outward when a was decreasing but a > 0. This was the same when a < 0 but the parabola was reflected over the x-axis. Finally when I moved the slider a while b = 1, the function s distance from each other changed. When a > 0, the distance between the two graphs increased as a increased. When a < 0, the same concept can be applied but the graphs are reflected over the y- axis.

3 Now I will move the c slider. When I moved this slider, all the graphs were translated up or down the y-axis. As c increases, the function translates up the y-axis. When c decreases, the function translates down the y-axis.

4 As you can see from all the graphs above, when I move the b slider, the composition of the graph changes. The composition of the graph means that the graph changes from linear to quadratic to cubic to root graphs. Some of the graphs have asymptotes, some have two separate graphs that do not connect, and others are only defined when the domain is x > Simultaneous Exploration Here I will explore 3 different cases where either a, b or c varies while the other two stay constant. 1. Case 1: b, c constant a varies: f x = a x! + c a = 3, 2, 1, 0, 1, 2, 3, b = 2, c = 0

5 When a varies, the width of the graph changes. When a > 1, the width of the graph shrinks as a increases. Then when 0 < a < 1, as a gets closer to zero, the width of the graph expands/gets wider. When a is positive, the range is [0, ) but when a is negative, the function is reflected over the x-axis and then the new range is (, 0]. The common point of all the graphs is (0,0). When a = 0, the graph is just a horizontal line along the x-axis. 2. Case 2: a, c constant b varies: f x = a x! + c a = 1, b = 3, 2, 1, 0, 1, 2, 3, c = 0 When b varies, the composition of the graph changes. The composition of the graph means that the graph changes from linear to quadratic to cubic to root graphs. There is a common point (1,1) that each graph satisfies. When b = 0, the graph is a linear horizontal line with the equation y = 1. Then when b = 1, there is a linear line with a slope of 1, which is represented by the green line.

6 3. Case 3: a, b constant c varies: f x = a x! + c a = 1, b = 3, c = 3, 2, 1, 0, 1, 2, 3 When c varies, the position of the graph on the y-axis changes. When c > 0, the graph will move upwards on the y-axis. When c < 0, the graph will shift down the y-axis. There is no common point in this set of graphs. When c = 0, the graph is at 0 on the y-axis. Part 2: Exploration of graphs of families of power functions with a given property 1. Roots Exploration Here I will overlay 15 graphs on the same axes with the property that they have an x- intercept in common. I used the parent graph f1 x = 3x! 3. The x-intercepts of this graph are (-1,0) and (1,0). I then manipulated the function in a few ways and discovered that in order to keep the x-intercepts the same, you need to vary the coefficient before x 2. I multiplied f1(x) by 14 different values and obtained the graphs below.

7 The roots of all of these graphs are x=1,-1 and the x-intercepts of all of the graphs are (- 1,0) and (1,0) just like the parent graph f1(x). All of the intersection points are at the x- intercepts or the roots of the graphs and it is the common points on all of the graphs. The one thing that I noticed about the orientation of the graph was the difference in width and the reflections over the x-axis. The common point of all the graphs is (0,0). When a = 0, the graph is just a horizontal line along the x-axis. When I multiplied by a number greater than 1, the width of the graph shrinks as the number I multiplied by increased. Then when the number I multiplied the parent graph was between 0 and 1, as a gets closer to zero, the width of the graph expands/gets wider. When the f1(x) was multiplied by a positive number, the range was [0, ) but when it was negative, the function is reflected over the x-axis and then the new range was (, 0]. Part 3: 3D plot of a graph of a power function 1. 3D graph Here I will make a 3D graph of the function f x = a x! + c with sliders for a, b, and c. I will manipulate the sliders in order to determine their effect on the graph. When I moved slider a while b = 1, the function s slope changed. When a > 0, the function will have a positive slope. When the positive a increases the slope of the function increases in steepness. When a < 0, the function has a negative slope and as the negative slope decreases, the slope increases in steepness. When I moved slider a while b = 2, the parabola s width changed. It shrunk when a was increasing and then it stretched outward when a was decreasing but a > 0. This was the same when a < 0 but the parabola was reflected over the x-axis.

8 Finally when I moved the slider a while b = 1, the function s distance from each other changed. When a > 0, the distance between the two graphs increased as a increased. When a < 0, the same concept can be applied but the graphs are reflected over the y- axis.

9 Now I will move the c slider. When I moved this slider, all the graphs were translated up or down the y-axis. As c increases, the function translates up the y-axis. When c decreases, the function translates down the y-axis.

10 II. IDP: As you can see from all the graphs above, when I move the b slider, the composition of the graph changes. Some have asymptotes, some have two separate graphs that do not connect, and others are only defined when the domain is x > 0. Content. Select content from the Common Core for Mathematics go to page 79 Describe: content here. (COMMON CORE STANDARDS) 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. Describe: Standards of mathematical Practice 2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. 8. Look for and express regularity in repeated reasoning. Pedagogy. Pedagogy includes both what the teacher does and what the student does. It includes where, what, and how learning takes place. It is about what works best for a particular content with the needs of the learner. 1. Describe instructional strategy (method) appropriate for the content, the learning environment, and students. This is what the teacher will plan and implement. This will be an exploratory lesson. Students will be put into pairs and use a software to assist them in discovering how a, b and c affect the graph. This will be useful so that students will not have to find the points to graph the function. They can use the parent graph and move it using the properties that they learned.

11 Include 21 st Century thinking skills: creative, critical, innovative, problem solving You may focus on just critical, or just creative, or both critical thinking and innovative problem solving. Technology. Digital tools using computers, Internet, and related technologies 2. Describe what learner will be able to do, say, write, calculate, or solve as the learning objective. This is what the student does. Students will be able to manipulate a parent function in order to fit the new function by using the properties of how a, b and c affect the graph. Students will be able to explain how a, b and c affects graphs of power relations. 3. Describe how 21 st Century skill is addressed. Students will think critically by using TI-Nspire CAS software sliders and discovering how the graph changes with each slider. Students will then use the properties discovered and use it on other functions. 1. Describe the technology The TI-Nspire CAS software is a computer version of the TI- Nspire CAS calculator. This calculator allows students to manipulate graphs and gather data and make graphs for the data. 2. Describe how the technology; i. Enhances the lesson, ii. Transforms content, and iii. Supports pedagogy. This software makes the lesson better because it allow students to manipulate a, b and c in a quadratic function without having to keep making multiple graphs in order to see how a, b and c affect the parabola. The software also transforms the content because instead of focusing our energy on making the graphs, students can shift their focus on how a, b and c affect the parabola. It also supports the pedagogy because students are able to view the affects of a, b and c on the parabola in real time. 3. Describe how the technology affects student s thinking processes. The technology affects the student s thinking processes by allowing them to move the sliders and watch how the graph changes. Then students are able to look at other graphs and either come up with the equation or graph the equation giving. This makes graphing or finding the equations of graphs easier and quicker. Reflect how did the lesson activity fit the content? How did the technology enhance both the content and the lesson Reflection: The lesson activity fits the content by having students discover the effects of certain constants by using sliders. The technology enhanced it because students were able to use the slider to move the function instead of having the students make different graphs

12 activity? showing the different shifts of the graphs. It is easier for the students to read and construct and observe by using the technology. III. Lesson Plan: Quadratic Exploration Learning Objectives: Students will be able to graph a quadratic equation by using the parent function and shifting it according to the properties learned. Essential Question: Can you graph a quadratic equation by shifting the parent function? Common Core State Mathematics Standards: 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. Common Core State Mathematical Practice Standards: MP 2 Reason abstractly and quantitatively. MP 5 Use appropriate tools strategically. MP 8 Look for and express regularity in repeated reasoning. Technology Standards: HS.TT.1.1 Use appropriate technology tools and other resources to access information HS.TT.1.2 Use appropriate technology tools and other resources to organize information HS.TT.1.3 Use appropriate technology tools and other resources to design products to share information with others Materials: TI-Nspire CAS calulators and software, pencil, paper Notes to Reader: Students have learned about quadratic equations and how to graph them by finding the x- intercepts and the vertex. Students have had little knowledge of how to create graphs and sliders on the TI-Nspire calculators. Time: 90 minutes

13 Phase 1: Problem Posing Phase 2: Small Group Investigation Phase 3: Whole Class Discussion Phase 4: Summarizing and Extending Students will graph a quadratic equation and will be timed. I will then ask students if there is a quicker way to graph quadratic equations. Students will give their ideas if they have any. If there are no ideas, I will ask students what the general form of a quadratic is. They will answer y = ax! + bx + c and I will ask them if a, b and c have a certain effect on the graph. Students will work on a pre-made TI-Nspire CAS document that has the general form of a quadratic function graphed with sliders for a, b and c. Students will move these sliders and observe how they affect the graph. Students will develop their conjectures about how a, b, and c affect the graph. Students will discuss their conjectures with the class. I will put up a quadratic function on the board and the class will discuss how to graphing it using their conjectures and we will check it by graphing it on the calculator. We will do a few more like this. Students will be timed again when graphing a quadratic equation by using the properties they just learned. Students should graph this quadratic quicker and will be asked which method of graphing that they find easier. Students will be assigned homework related to the classwork. IV. Reflection The use of technology to do this exploration made it a lot easier. If this were to be done by hand, it would be harder to graph all the equations on one graph and color-code them where they are easily seen and shows how they are different. Also, if this were done by hand, it would take forever. This allowed me to focus more on how a, b and c were changing the graphs instead of focusing on making the graphs by hand, which helped my understanding in the end. I also liked how the software was able to type in one equation and have one coefficient be multiple numbers and it will graph all of them instead of having to input them separately. It was definitely a time saver. TI-Nspire CAS software will affect the students understanding by allowing them to explore the affects of a, b and c on a graph by just sliding a button. It is so easy to see the changes in the graph using it. It is time saving and more engaging than having to do this one paper, which would become very tedious after a while. Students will be able to see how a, b and c affect the graphs in real time and it will be visually engaging and appealing to them. They are able to solely focus on the changes in the graph and spend more time working on manipulating parent graphs using their new knowledge about how a, b and c affect the graph instead of focusing and spending most of their time on constructing the graphs.

Algebra II Notes Transformations Unit 1.1. Math Background

Algebra II Notes Transformations Unit 1.1. Math Background Lesson. - Parent Functions and Transformations Math Background Previously, you Studied linear, absolute value, exponential and quadratic equations Graphed linear, absolute value, exponential and quadratic