NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M4 ALGEBRA I Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions Student Outcomes . Students compare the basic quadratic (parent) function, 푦 = 푥2, to the square root function and do the same with cubic and cube root functions. They then sketch graphs of square root and cube root functions, taking into consideration any constraints on the domain and range. Lesson Notes In these exercises, students explore the effects of squaring, taking the square root of, cubing, and taking the cube root of various values. They then use visible patterns to make generalizations about the graphs of square root and cube root functions, as well as cubic functions. Classwork Opening Exercise (5 minutes) Students review evaluating expressions that involve radicals and exponents so that they Scaffolding: are prepared to work with quadratic, square root, cubic, and cube root functions. For students who are not as familiar working with radicals, Opening Exercise it may be important to spend a a. Evaluate 풙ퟐ when 풙 = ퟕ. few minutes discussing the ퟒퟗ difference between the solutions for the equations 푥2 = 25 and 푥 = √25. In the b. Evaluate 풙 when 풙 = ퟖퟏ. √ first case, there are two ퟗ possible solutions: 5 and −5. But in the second, there is only c. Evaluate 풙ퟑ when 풙 = ퟓ. one: 5. The square root symbol originates from the ퟏퟐퟓ geometric application of finding the length of the hypotenuse. ퟑ d. Evaluate √풙 when 풙 = ퟐퟕ. The use of that symbol is ퟑ reserved to mean the positive value that when squared would yield the value inside. Hence, when we solve for 푥 in the equation 푥2 = 25, we state the solution as 푥 = ±√25. Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions 200 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M4 ALGEBRA I Exploratory Challenge 1 (5 minutes) Exploratory Challenge 1 Use your graphing calculator to create a data table for the functions 풚 = 풙ퟐ and 풚 = √풙 for a variety of 풙-values. Use both negative and positive numbers, and round decimal answers to the nearest hundredth. 풙 풚 = 풙ퟐ 풚 = √풙 Scaffolding: ퟒ ퟏퟔ ퟐ . Provide struggling ퟐ ퟒ ퟏ. ퟒퟏ students with tables that ퟎ ퟎ ퟎ include several 푥-values. −ퟐ ퟒ Error . Some students might need −ퟒ ퟏퟔ Error to try many more values before recognizing the patterns. Discussion (5 minutes) Compare the two 푦-columns in the data table from Exercise 1. Students can use their calculators to explore additional values of these functions. Observe both 푦-columns from above. What do you notice about the values in the two 푦-columns? In the column for 푦 = 푥2, all 푦-values are positive. In the column for 푦 = √푥, all negative 푥-values produce an error. Why are all 푦-values for 푦 = 푥2 positive? All 푦-values are positive since they are obtained by squaring the 푥-value. Why do all negative 푥-values produce an error for 푦 = √푥? No real number, when squared, produces a negative result. Therefore, the calculator produces an error. What is the domain of 푦 = 푥2 and 푦 = √푥? The domain of 푦 = 푥2 is all real numbers. The domain of 푦 = √푥 is 푥 ≥ 0. What is the range of 푦 = 푥2 and 푦 = √푥? The range of 푦 = 푥2 is 푦 ≥ 0. The range of 푦 = √푥 is 푦 ≥ 0. Compare the domain and range of 푦 = 푥2 and 푦 = √푥. The domain of 푦 = √푥 is limited to nonnegative values, while the domain of 푦 = 푥2 includes all real numbers. The range of both functions is the same set of all nonnegative real numbers. What is the result if we take the square root of 푥2? Have students try making a third column in the chart to see if they can come up with a rule for √푥2. This should help them understand the need for the ± when taking the square root of a variable expression. If 푥 is a negative number, the result is (−푥), and if 푥 is a positive number, the result is 푥. So, √푥2 = |푥|. Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions 201 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M4 ALGEBRA I Exploratory Challenge 2 (10 minutes) Exploratory Challenge 2 Scaffolding: Create the graphs of 풚 = 풙ퟐ and 풚 = √풙 on the same set of axes. Provide students with the axes drawn and numbered. What additional observations can we make when comparing the graphs of these functions? They intersect at (0, 0) and (1, 1). The square root function is a reflection of the part of the quadratic function in the first quadrant, about 푦 = 푥 (when 푥 is nonnegative). Why do they intersect at (0, 0) and (1, 1)? 02 and √0 are both equal to 0. 12 and √1 are both equal to 1. Exploratory Challenge 3 (15 minutes) . Predict the relationship between 푦 = 푥3 and 푦 = 3√푥. Both functions include all real numbers in their domain and range since a cubed number can be positive or negative, as well as the cube root of a number. Exploratory Challenge 3 Create a data table for 풚 = 풙ퟑ and 풚 = ퟑ√풙, and graph both functions on the same set of axes. Round decimal answers to the nearest hundredth. 풙 풚 = 풙ퟑ 풚 = ퟑ√풙 −ퟖ −ퟓퟏퟐ −ퟐ −ퟐ −ퟖ −ퟏ. ퟐퟔ −ퟏ −ퟏ −ퟏ ퟎ ퟎ ퟎ ퟏ ퟏ ퟏ ퟐ ퟖ ퟏ. ퟐퟔ ퟖ ퟓퟏퟐ ퟐ Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions 202 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M4 ALGEBRA I . Using the table and graphs, what observations can you make about the relationships between 푦 = 푥3 and 푦 = 3√푥? They share the points (0, 0), (1, 1), and (−1, −1). The domain and range of both functions are all real numbers. The functions are symmetrical about the origin. Each of these two functions is a reflection of the other across the line 푦 = 푥. If students do not arrive at the responses above, use the following prompts: . What are the domain and range of each function? . How are the graphs related to each other? . Describe the symmetry that exists within the tables and graphs. Closing (2 minutes) . The square root function is a reflection of the quadratic function across the line 푦 = 푥, when 푥 is nonnegative. The domain of 푦 = 푥2, 푦 = 푥3, and 푦 = 3√푥 is all real numbers. The domain of 푦 = √푥 is 푥 ≥ 0. The range of 푦 = 푥2 and 푦 = √푥 is 푦 ≥ 0. The range of 푦 = 푥3 and 푦 = 3√푥 is all real numbers. 푦 = 푥3 and 푦 = 3√푥 are each symmetrical about the origin and are reflections of each other across the line 푦 = 푥; the two operations reverse each other. Note that inverse functions have not yet been introduced, but this is an opportunity to offer a preview, depending on the ability and interest level of your students. Lesson Summary . The square root parent function is a reflection of the quadratic parent function across the line 풚 = 풙, when 풙 is nonnegative. The domain of quadratic, cubic, and cube root parent functions is all real numbers. The domain of the square root parent function is 풙 ≥ ퟎ. The range of quadratic and square root parent functions is [ퟎ, ∞). The range of the cubic and cube root parent functions is all real numbers. The cube root and cubic parent functions are symmetrical about the origin and are reflections of each other across the line 풚 = 풙; the two operations reverse each other. Exit Ticket (3 minutes) Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions 203 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M4 ALGEBRA I Name Date Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions Exit Ticket 1. Describe the relationship between the graphs of 푦 = 푥2 and 푦 = √푥. How are they alike? How are they different? 2. Describe the relationship between the graphs of 푦 = 푥3 and 푦 = 3√푥. How are they alike? How are they different? Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions 204 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M4 ALGEBRA I Exit Ticket Sample Solutions 1.