Algebra 2 Honors Lesson 10 Translating Functions
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1 Algebra 2 Honors Lesson 10 Translating Functions Objectives: The students will be able to translate a base function horizontally and vertically. Students will be able to describe the translation of f(x) to g(x), given that g(x) = f(x h) + k. Students will determine that the vertex of f(x) = (x h) 2 + k is (h, k). Materials: Hw #2-9 answers overhead; tally sheets; Translating Functions handouts and overheads; note-taking templates; homework #2-10: translation practice Time Activity 10 min Homework Review Students check their answers to hw #2-9 and discuss work with their group. Pass around a tally sheet for questions (one from each side of the room to speed it up). 5 min Homework Presentations Review the top 2 or 3 questions from the tally sheet. Problems to grade: 20 min Do Now Hand out the Translating Functions worksheet to students. Problem 1 has two parabolas, where the second is a vertical translation of the first. Problem 2 has two parabolas also, where the second is a horizontal translation of the first. Ask students to do these, but to skip the calculator parts for now. 25 min Group Work Show the graphs from parts 1 and 2 on the overhead and take any questions. Now, students should work on the calculator portions of the translations worksheet. 20 min Direct Instruction Lesson Name: Translating Functions Portfolio Section: Functions Background: To translate in math means to move a graph from one place to another. Concepts: Vertical Shift - The graph of y = f(x) + k is the same as the graph of y = f(x), except shifted vertically by k steps. - If k < 0, then f(x) is shifted down. - If k > 0, then f(x) is shifted up. Horizontal Shift - The graph of y = f(x h) is the same as the graph of y = f(x), except shifted horizontally by h steps. - If h < 0, then f(x) is shifted right. - If k > 0, then f(x) is shifted left. Note that this seems opposite of what you d expect! For any function - y = f(x h) + k is the exact same shape as y = f(x), except shifted h spaces horizontally and k spaces vertically. Parabola (Vertex Form) - The function f(x) = (x h) 2 + k is a translation of f(x) = x 2. o Shifted h spaces horizontally and k spaces vertically. o The vertex is at (h, k) Example: Graph the function f(x) = (x + 3) 2 6 Start by finding the vertex: (-3, -6). Sketch a set of axes and plot the point. Plug in a few x-values to one side of the -3, noting that the symmetry of a parabola allows you to use only one side. Homework #2-10: Translation Practice
2 Algebra 2 Honors *Lesson #2-10 Name: Translating Functions Part 1: Vertical Shifts Fill in the table for f(x) and g(x). Then, graph each function on the axes. x f(x) = x 2 g(x) = x How do the g(x) values compare to the f(x) values? Calculator Part 1: Set your calculator window to the following: -6 < x < 6 and -3 < y < 6 Graph all of the following functions on the same screen. After you put in each new function, view the graph to see what it looks like. 1) Y 1 x 2 3) Y 3 x 2 2 5) Y 5 x 2 2 2) Y 2 x 2 1 4) Y 4 x 2 1 1) Based on your results, describe how adding or subtracting a number to f(x) = x 2 affects the graph. 2) Predict: what will the graph of f(x) = x look like? Check it on your calculator (you will need to change your viewing window).
3 Part 2: Horizontal Shifts Fill in the table for f(x) and g(x). Then, graph each function on the axes. x f(x) = x 2 g(x) = (x 2) How do the g(x) values compare to the f(x) values? Calculator Part 2: Set your calculator window to the following: -6 < x < 6 and -3 < y < 6 Graph all of the following functions on the same screen. After you put in each new function, view the graph to see what it looks like. 1) Y 1 x 2 3) Y 2 x 3 2) Y 2 x 1 2 4) Y 2 x ) Y 2 x 4 2 3) Based on your results, describe how adding or subtracting a number to the x before squaring affects the graph of f(x) = x 2. Is there anything surprising about it? 4) Predict: what will the graph of f(x) = (x + 7) 2 look like? Check it on your calculator (you will need to change your viewing window). 5) Synthesize: based on your results in part 1 and in part 2, what do you think the graph of f (x) x will look like? Graph it on your calculator to check. 6) Synthesize: Given that the vertex of f (x) x 2 is (0, 0), what will be the vertex of 2 k? f (x) x h
4 Algebra 2 Honors Lesson #2-10 Name: Homework #2-10: Translating Functions 1) Write the function whose graph is the same as the graph of y = x 2, but is translated: a. to the right 4 units. b. down 3 units. c. to the left 7.5 units. d. up units. e. to the right 3 and down 7 units. f. to the left 8 and up 8 units. 2) If (0, 3) is a point on the graph of y = f(x), which of the following points must be on the graph of y = f(x) + 5? a. (0, -2) b. (5, 3) c. (0, 8) d. (-5, 3) 3) If (7, 3) is a point on the graph of y = f(x), which of the following points must be on the graph of y = f(x + 3)? a. (4, 3) b. (11, 3) c. (7, 6) d. (7, 0) 4) If (-2, 8) is a point on the graph of y = f(x), which of the following points must be on the graph of y = f(x 2) + 4? a. (-4, 12) b. (-4, 4) c. (0, 4) d. (0, 12) 5) Write the function of each parabola on the graph. They are all translations of f(x) = x 2. a(x) = b(x) = c(x) = d(x) =
5 6) You are given the graph of y = f(x) below. On the same coordinate plane, graph the functions g(x), h(x), s(x), and t(x) using f(x) as a base. Use different colors for the different functions. g(x) = f(x) + 5 h(x) = f(x + 3) s(x) = f(x 4) t(x) = f(x 3) 6 7) Make a graph of each parabola. Do this by first finding and plotting the vertex, and then plug in a few x-values on one side of the vertex. Use symmetry to find points on the other side. 8) f (x) x g(x) x Vertex: (, ) Vertex: (, ) Hint: pick a point on the original function, and determine how it has moved to get to its new location. a. b. c. d.
6 Hw #2-9 Tally Sheet Page 1 1) 2) 3) 4) Page 2 Graph Problem) 1) 2) 3)
7 Hw #2-9 Answers 1) f (x) 3) f (x) Page 2: 2x 6, x 1 2x 6, x 1 6, 5 x 3 3x 12, x 3 2) f (x) 2x 5, x 0 0.5, 0 x 3 2 x 1, x 3 3 2x 16, x 3 4) f (x) 5, 3 x 2 4x 6, x 2 1) x = -1/3 or x = 5 2) -17 < x < 7 3) x < -3 or x > 8 Bonus: The lines intersect at (7, 25.5)
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