Download presentation
Presentation is loading. Please wait.
Published byAmice Chandler Modified over 6 years ago
1
1.2A Stretches The graph of y + 3 = f(x) is the graph of f(x) translated… up 3 units left 3 units down 3 units right 3 units x 2. The graph of f(x) + 4 is the graph of f(x) translated… 4 units up 4 units left 4 units down 4 units right x 3. The graph of f(x – 7) + 6 is the graph of f(x) that has been translated.. 7 units left, 6 units up 7 units left, 6 units down 7 units right, 6 units down 7 units right, 6 units up x 4. In general, the graph of f(x – h) + k, where h and k are positive, as compared to the parent function graph f(x), is translated h units left and k units up h units right and k units up h units left and k units down h units right and k units down x Math 30-1
2
5. When the output of a function y = f(x) is multiplied by -1, the result, y = -f(x), is a reflection of the graph in the x – axis y – axis line y = x. 6. When y = f(x) is transformed to y = f(-x), then (x, y) is transformed to (-x, -y) (x , -y) (-x, y) Math 30-1
3
Vertical Stretches: The effect of parameter a.
A stretch changes the shape of the graph. (Translations changed the position of a graph.) Function Transformations: Vertical Stretch |a| describes a vertical stretch about the x-axis. An Invariant point is a point on a graph that remains unchanged after a transformation. 1.2A Vertical Stretches Math 30-1
4
Vertical Stretches In general, for any function y = f(x), the graph of the function y =a f(x) has been vertically stretched about the x-axis by a factor of |a| . The point (x, y) → (x, ay). Only the y coordinates are affected. Invariant points are on the line of stretch, the x-axis. are the x-intercepts. When |a| > 1, the points on the graph move farther away from the x-axis. Vertical stretch by a factor of 3 When |a|< 1, the points on the graph move closer to the x-axis. Vertical stretch by a factor of ⅓ Math 30-1
5
Vertical Stretching y = af(x), |a| > 1
y = 2f(x) A vertical stretch about the x-axis by a factor of 2. y = f(x) Key Points (x, y) → (x, 2y) (-2, 0) → (-2, 0) (-1, -7) → (-1, -14) (1/2, 1) → (1/2, 2) Invariant Points Domain and Range (-2, 0) (0, 0) (1, 0) (2, 0) Math 30-1
6
Vertical Stretches about the x-axis y = af(x), |a| < 1
Consider Write the equation of the function after a vertical stretch about the x-axis by a factor of ½. Write the transformation in function notation. Write the coordinates of the image of the point (-7, 7) (-7, 3.5) The point (x, y) maps to List any invariant points. How are the domain and range affected? Math 30-1
7
The graph of g(x) is a transformation of f(x).
Horizontal stretch By a factor of ½ (2, 0)→ (1, 0) (x, y)→(½x, y) Invariant Point On the y-axis Is the transformation a translation? Is the transformation a vertical stretch? Math 30-1
8
Horizontal Stretches In general, for any function y = f(x), the graph of the function y = f(bx) has been horizontally stretched by a factor of The point (x, y) → Only the x coordinates are affected. Invariant points are on the line of stretch, the y-axis. are the y-intercepts. When |b| > 1, the points on the graph move closer to the y-axis. Horizontal stretch by a factor of ⅓ When |b|< 1, the points on the graph move farther away from the x-axis. Horizontal stretch by a factor of 4 Math 30-1
9
Characteristics of Horizontal Stretches about the y-axis
Consider Write the equation of the function after a horizontal stretch about the y-axis by a factor of 3. Write the transformation in function notation. Can it be written in any other way? Write the coordinates of the image of the point (-3, 3 ) → (-9, 3) Write the coordinates of the image of the point (x, y) List any invariant points. How are the domain and range affected? Math 30-1
10
Is This a Horizontal or Vertical Stretch of y = f(x)?
The graph y = f(x) is stretched vertically about the x-axis by a factor of Math 30-1
11
Has the graph been stretched Horizontally or Vertically?
(1, 1) Math 30-1
12
Zeros of a Function 1. What are the zeros of the function?
x = -2, 0, 3 2. Use transformations to determine the zeros of the following functions. x = -1, 1, 4 x = -2, 0, 3 x = -4, 0, 6 Math 30-1
13
Describing the Horizontal or Vertical Stretch of a Function
Describe the transformation in words compared to the graph of a function y = f(x). a) y = f(3x) b) 3y = f(x) Stretched horizontally by a factor of about the y-axis. Stretch vertically by a factor of about the x-axis. c) y = f( x) d) y = 2f(x) Stretched vertically by a factor of about the x-axis. Stretched horizontally by a factor of 2 about the y-axis. Math 30-1
14
Consider the graph of a function y = f(x)
The transformation described by y = f(2x+4) is horizontal stretch about the y-axis by a factor of ½. The translation described by y = f(2x + 4) is horizontal shift of 4 units to the left. y = f(2(x + 2)) Translations must be factored Math 30-1
15
Stating the Equation of y = af(kx)
The graph of the function y = f(x) is transformed as described. Write the new equation in the form y = af(bx). Stretched horizontal by a factor of one-third about the y-axis, and stretched vertically about the x-axis by a factor of two. y = 2f(3x) Stretched horizontally by a factor of two about the y-axis and translated four units to the left. Math 30-1
16
Assignment Page 28 2, 5a,b, 6, 7a,c, 8, 13, 14c, d Math 30-1
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.