M.I. Transformations of Functions

Transcription

1 M.I. Transformations of Functions Do Now: A parabola with equation y = (x 3) is translated. The image of the parabola after the translation has an equation of y = (x + 5) 2 4. Describe the movement. 1

2 Translations A vertical translation of the graph of y = f(x) is a shift of the graph up or down in the coordinate plane. Graph the following functions: y 1 = x 2 y 2 = x 2 5 y 3 = x What effect does the 5 and the + 2 have on x 2? Does the same hold true for other functions? y 1 = x y 2 = x 4 y 3 = x + 3 y 1 = y 2 = y 3 = y 1 = y 2 = y 3 = A vertical translation of y = f(x) is of the form: 2

3 A horizontal translationof the graph of y = f(x) is a shift of the graph left or right in the coordinate plane. Graph the following functions: y 1 = x 2 y 2 = (x 5) 2 y 3 = (x + 2) 2 What effect does the 5 and the + 2 have on x 2? y 1 = x y 2 = x 4 y 3 = x + 3 y 1 = x y 2 = (x 3) y 3 = (x + 6) y 1 = y 2 = y 3 = y 1 = y 2 = y 3 = A horizontal translation of y = f(x) is of the form: 3

4 Given the parent function, determine the function for each after the given transformation. 1. left 3 2. up 7 3. down 9 4. right 6 and down 2 5. up 1 and left 5 4

5 Use transformations of the graphs of and to sketch 5

6 Write separate functions that translate the graph of the parent functions and a) down 4 units b) up 2 units c) left 7 units d) right 1 unit e) up 5 units and left 2 units. f) right 4 units and down 3 units. 6

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8 M.I. Transformations (Day 3) Reflections across axes How have you been taught to reflect a point over the x axis? over the y axis? r x axis (x, y) : r y axis (x, y) : Graph the following function: y 1 = (x + 3) 2 4 To reflect over the x axis, negate the y...or the entire y function. y 2 = [(x + 3) 2 4] To reflect over the y axis, negate the x...or all x values. y 2 = ( x + 3) 2 4 y 1 = x 2 + 4x + 2 y 2 = ( x) 2 + 4( x) + 2 8

9 Given the equation y = x 2 + 8x + 2, determine the equation: a) after reflection over y axis b) after reflection over x axis To determine the equation after a translation, lets identify the vertex first. To do this, we must convert from standard form to vertex form... y = x 2 + 8x + 2 Turning Point/Vertex: To translate this equation, translate the turning point. Then write the new equation in vertex form. c) after translation of 6 units up and 2 units left. b) after T <5, 1> 9

10 Graph y = x 2 "Stretches and Shrinks" (2, 4) Multiplying the function by a constant greater than 1 will cause the function to... For example... y = 2x 2 (2, 8) (2, 4) Multiplying the function by a constant between 0 and 1 will cause the function to... For example... 1 y = x 3 2 (3, 9) (3, 9) (3, 3) In general, let c be a real number not equal to 0. If f(x) is known, then g(x) = cf(x) is a vertical stretch if c > 1, and a vertical shrink if 0 < c < 1 Vertical stretches/shrinks: x values remain the same y values are multiplied by c for a stretch y values are mulitplied by 1/c for a shrink Practice: Write the equations of y 1, a vertical stretch of y by the factor of 3 and y 2, a vertical shrink of 1/3. Ex 1: Ex 2: Ex 3: Describe how y = x 2 can be transformed to... 10

11 To stretch or shrink the graph in the x direction (horizontally), multiply the input, x, by a constant... y = sin(x) (90 o, 1) y = sin(2x) has a period of (45 o, 1) (90 o, 1) (here x values need to be halved to have the same output) Given a function f(x), we obtain f(dx), where d is a constant, by shrinking/compressing the graph horizontally by a factor of (1/d). That is, we leave the y values the same, and multiply all x values by (1/d). Horizontal Shrinks/Compressions: y values remain the same all x values are multiplied by a horizontal shrink factor of 1/d Practice: Write the equation of y 1, a horizontal shrink of y by the factor of 1/4. Ex 1: Ex 2: Ex 3: What is the horizontal shrink factor of y = 2x 2 11

12 y = sin(x) (90 o, 1) 1 y = sin( x) 2 has a period of (180 o, 1) (here x values need to be doubled to have the same output) Given a function f(x), we obtain f((1/d)x), where d is a constant, by stretching the graph horizontally by a factor of (d). That is, we leave the y values the same, and multiply all x values by d. Horizonal Stretches: y values remain the same all x values are multiplied by a horizontal stretch factor of d Practice: Write the equation of y 1, a horizontal stretch of y by the factor of 3. Ex 1: Ex 2: Ex 3: What is the horizontal stretch factor of 12

13 In general, let d be a real number not equal to 0. If f(x) is known, then g(x) = f(dx) is a horizontal stretch/ compression if d > 1, and a horizontal shrink if 0 < d < 1 1. Find the equation of the parabola formed by stretching vertically by a factor of Find the equation of the parabola formed by compressing vertically by a factor of 1/2. 3. Find the equation of the parabola formed by stretching horizontally by a factor of Find the equation of the parabola formed by shrinking/compressing horizontally by a factor of 1/3. 13

14 Given the graph of transformation to:, describe the a) b) c) d) 14