Relating Quadratic Functions to Graphs

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1 Relating Quadratic Functions to Graphs Student Probe Explain the change from: a. b. c. The change from g x x h x x j x x 3 g x 4 x is the parabola becomes narrower, containing the point 1, rather than 1,1. The changes from h x x 4 are that the parabola is flipped or upside down and the vertex is 0,4 rather than 0,0. The change from j x x 3 is the parabola has a horizontal translation to the right. The vertex becomes 0,3 rather than 0,0. Lesson Description This lesson uses the concept of transformation of functions to relate graphs of the basic quadratic function graphs of functions of the form ax, ax c, and x b. Rationale Understanding functions and the relationships among their equations, data tables, and graphs is important in developing mathematical fluency. Multiple representations of functions help students make connections and think in mathematically flexible ways. Quadratic functions describe a variety of real world situations, including projectile motion. Students who can effectively interpret quadratic functions will be successful in scientific, as well as mathematical applications. At a Glance What: Transformations of quadratic functions. Common Core State Standard: CC.9-1.F.BF.3 Build new functions from existing functions. 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. Matched Arkansas Standard: AR.9-1.RF.AII.1.4 (RF.1.AII.4) Analyze and report, with and without appropriate technology, the effect of changing coefficients, exponents, and other parameters on functions and their graphs (linear, quadratic, and higher degree polynomial) Mathematical Practices: Make sense of problems and persevere in solving them. Who: Students who have difficulty relating a function to its graph. Grade Level: Algebra Prerequisite Vocabulary: transformation, quadratic, coefficient, minimum, maximum, y-intercept, parent graph, x-intercept, parabola, vertex Prerequisite Skills: graphing, use of a graphing calculator, building a table of values Delivery Format: Individual, small group, whole group Lesson Length: minutes Materials, Resources, Technology: graph paper, colored pencils or markers, graphing calculator Student Worksheets: None

Preparation Students should be provided with graph paper with the axes labeled and colored pencils. It is strongly suggested that students have access to graphing calculators during this lesson.

2 Preparation Students should be provided with graph paper with the axes labeled and colored pencils. It is strongly suggested that students have access to graphing calculators during this lesson. Lesson The Expect students to say or do If students do not, then the 1. Here is the quadratic function x Since this is the simplest form of a quadratic function, we call this parabola the parent function.. We will be referring to this graph often. Describe what you notice about it. 3. Let s try another function: g x x. Predict what you think the graph might look like. (Note: many students will think the graph will be fatter.) Answers will vary, but listen for, it is U-shaped, it opens up, It contains the points 1,1, 0,0, and 1,1, the vertex is 0,0. Answers will vary, but listen for U-shaped, opens up. fatter, skinnier. What shape is it? Does it open up or down? What are some of the points on the graph? What is the vertex? Do you think it will be fatter or skinnier?

The Expect students to say or do If students do not, then the 4. Graph the function You may use two different g x x on the same colors to graph the two axes. functions.

How is the equation for different from the equation for g x? 6. How do you think we could make the graph of a parabola wider?

3 The Expect students to say or do If students do not, then the 4. Graph the function You may use two different g x x on the same colors to graph the two axes. functions. How is it like the parent function? How is it different from the parent function? 5. How is the equation for like the equation for g x? How is the equation for different from the equation for g x? 6. How do you think we could make the graph of a parabola wider? Students should see that the graph of g x is narrower or skinnier than the graph of. The coefficient changed from 1 in to in g x, Or The number in front o is 1 in and in g x. Accept any answer in which the coefficient is changed to between 1 and 0. 1 Example: x. 4 graph students predictions. Why don t we try 1 x? 4

The Expect students to say or do If students do not, then the 7. What can we conclude If a 1, the graph is about the graphs of normal. graph a variety of functions. ax? If a 1, the graph is skinny.

Predict what you think the graph might look like. Graph and h x axes. x on the same 9. How is the graph of h x like the graph of the parent function?

4 The Expect students to say or do If students do not, then the 7. What can we conclude If a 1, the graph is about the graphs of normal. graph a variety of functions. ax? If a 1, the graph is skinny. Have students predict (Note: a 0 will be If 0 a 1, the graph is fat. normal, fat, or skinny addressed in the next before they see the graph. step.) 8. Now let s try the function Refer to Step. h x x. Predict what you think the graph might look like. Graph and h x axes. x on the same 9. How is the graph of h x like the graph of the parent function? How is the graph of h x different from the graph of the parent function? 10. How is the equation for like the equation for h x? How is the equation for different from the equation for h x? 11. What can we conclude about the graphs of ax when a 0? Reflected (flipped) over the x- axis Instead of opening up the graph opens down has a minimum and h x has a maximum. The coefficient for is 1 and the coefficient for h x is 1. The graph will be flipped. Use different colors for each graph. graph a variety of functions. Have students predict whether the graph will be flipped or not.

The Expect students to say or do If students do not, then the 1. Here is another function: j x x 4. Predict what you think the graph might look like. Graph x and j x x 4 on the same axes. 13.

How would you change the function j x x to make the graph shift down 4 units (or change the y-intercept to 0, 4? 4 It shifted up 4 units The y-intercept changed from 0,0 to 0,4. j x has a positive 4.

5 The Expect students to say or do If students do not, then the 1. Here is another function: j x x 4. Predict what you think the graph might look like. Graph x and j x x 4 on the same axes. 13. How are the graphs alike? How are the graphs different? 14. How are the equations alike? How are the equations different? 15. How would you change the function j x x to make the graph shift down 4 units (or change the y-intercept to 0, 4? 4 It shifted up 4 units The y-intercept changed from 0,0 to 0,4. j x has a positive 4. Change 4 to 4. Or j x x 4 graph students predictions. Why don t we try j x x 4? 16. What can we conclude about x c? If c 0, the parabola shifts up. If c 0, the parabola shifts down. graph a variety of functions. Have students predict whether the graph will be shifted up or down.

The Expect students to say or do If students do not, then the 17. Here is another function k x x 3. Predict what you think the graph might look like. Graph x and k x x 3 on the same axes. 18.

6 The Expect students to say or do If students do not, then the 17. Here is another function k x x 3. Predict what you think the graph might look like. Graph x and k x x 3 on the same axes. 18. How are the graphs alike? How are they different? 19. How are the equations for and k x alike? How are they different? 0. How would you change the function move it 3 units to the left? 1. What can we conclude about x b? It moved to the right 3 units. It slid 3 units to the right on the x-axis. The x-intercept changed to 3,0. In k x, you subtract 3 from x inside the parenthesis. x 3 If b 0, the parabola shifts to the right. If b 0, the parabola shifts to the left. graph students predictions. Why don t we try x 3? graph a variety of functions. Have students predict whether the graph will shift left or right.

7 The Expect students to say or do If students do not, then the. Now let s put this all together. Without graphing, predict what the graph of this function will look like: Answers will vary, but listen for, fat, opens up, shifted left units, shifted up 1 unit. Will it be fat, normal, or skinny? Will it be shifted to the right or to the left? How far? 1 Will it be shifted up or down? x 1 3 How far? What is the vertex? The vertex is,1. Graph the function to check your predictions. If 0,0 is the normal vertex and we shifted left and up 1, what is this vertex? Teacher Notes 1. It is strongly recommended that students have access to graphing calculators for this lesson.. Have students predict what they think the graph of a transformed equation will look like, and then test their hypothesis by graphing. 3. Students may benefit from generating a data table and graphing a few functions by hand. 4. Equations of the form x b can be confusing for students. Additional time may be required for functions of this form. 5. It may be beneficial to spread this lesson over several days, addressing each parameter change in its own lesson. In that case, be sure to allow an additional day for addressing all the parameter changes in a single function (such as Step.) Variations None Formative Assessment 1 Describe how the graph of g x x 3differs from the graph of x. Answer: g x is flipped, skinny, shifted units to the right, and shifted down 3 units. Its vertex is, 3 rather than 0,0.

8 References Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in Mathematics. Retrieved June 1, 011, from rti4sucess. Paulsen, K., & the IRIS Center. (n.d.). Algebra (part ): Applying learning strategies to intermediate algebra. Retrieved on June 1, 011.

Completing the Square

Completing the Square Student Probe Write this expression in vertex form:. Answer: Lesson Description This lesson uses an area model of multiplication and algebra tiles to develop the concept of completing