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Chapter 3. Transformations of Graphs and Data. Transformation: 1 to 1 correspondence between sets of points (augmenting a figure) Translation: slide Scale change: shrinking or stretching a figure Translations and scale changes are two types of transformations (there are others).

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Chapter 3

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chapter 3

Chapter 3

Transformations of Graphs and Data


Transformation: 1 to 1 correspondence between sets of points (augmenting a figure)

  • Translation: slide
  • Scale change: shrinking or stretching a figure
  • Translations and scale changes are two types of transformations (there are others)
section 3 1 changing windows
Section 3-1 Changing Windows
  • To get your calculator back to the default window: Hit the Zoom button and then chose 6: Zstandard
  • The default window is:
    • -10 < x < 10 and -10 < y < 10
  • There are parent functions that you must become familiar with (see Appendix A on page 162)
    • Floor function, hyperbola, cubic, inverse-square curve, square root, absolute value function, exponential function, etc.

Asymptote: place on a graph where the graph skips (where it is discontinuous)

  • i.e. the x and y axes are asymptotes for hyperbolas and inverse-square curves
  • Sometimes your graphing calculator will draw in the asymptotes, even though they are not a part of the final graph
  • Read page 164: criteria for sketches (and then follow the criteria)
section 3 2 the graph translation theorem
Section 3-2 The Graph-Translation Theorem
  • The original function is called the preimage, the function after the transformation is the image
  • There are three different ways to write translations:
    • 1) (x, y) → (x + h, y + k)
    • 2) T(x, y) = (x + h, y + k)
    • 3) Th,k

The value added to the x value shifts the graph horizontally and the value added to the y value shifts the graph vertically

  • When you write the translation in the equation, it looks like it is moving in the opposite direction
  • i.e. if the preimage is y = x² and the image is y = (x + 2)², that is a horizontal shift 2 places to the left
  • Carefully reread the Graph-Translation Theorem

Under the translation T(x,y) = (x + h, y + k) the image of y = f(x) is y – k = f(x – h)

  • Typically we move k over to the right side of the equation
  • Ex1. If the preimage is and the image is describe the translation
section 3 4 symmetries of graphs
Section 3-4 Symmetries of Graphs
  • A figure is said to be reflection-symmetric if it can be mapped onto itself by reflection over a line (explained as “reflectional symmetry over the line __________”)
  • That line is called the axis of symmetry or the line of symmetry
  • If a figure has 180° rotational symmetry it is said that it has symmetry to point P or point symmetry
  • P is called the center of symmetry

Parabolas have reflectional symmetry and cubics have point symmetry

  • A figure is symmetric to the origin if it has 180° rotational symmetry around the origin, like hyperbolas
  • A figure is symmetric with respect to the x-axis if it has reflectional symmetry over the x-axis (i.e. x = y²)
  • A power function is one in which the independent variable is raised to a power greater than or equal to 2

An even function has reflectional symmetry over the y-axis

  • An odd function has 180° rotational symmetry around the origin
  • To prove a function odd or even you must use the algebraic definitions
    • Even: f(x) = f(-x) and Odd: f(-x) = -f(x)
    • See example 3 on page 181
  • If f is a function and (x,y) → (x + h, y + k), then all lines of symmetry are mapped as well
section 3 5 the graph scale change theorem
Section 3-5 The Graph Scale Change Theorem
  • Scale changes stretch or shrink the figure so that the image and preimage are no longer congruent
  • There are 3 ways to write scale changes
    • S(x,y) = (ax, by)
    • S:(x,y) → (ax, by)
    • Sa,b
  • The horizontal stretch is a and the vertical stretch is b

Like with translations, scale changes look “opposite” in the equation

  • i.e. if the preimage is y = x² and the image is y = (3x)², the scale change is a horizontal shrink of magnitude ⅓
  • If a scale change has the same magnitude vertically as it has horizontally, then it is a size change
  • Reread and study the graph scale-change theorem (page 188)

S(x,y) = (ax, by)

    • If a is negative, then the graph is reflected over the y-axis
    • If b is negative, then the graph is reflected over the x-axis
  • Ex1. f(x) = x³ - 4x and g(x) = 3x³ - 12x write the scale change
  • Ex2. If f(x) = x², write the equation for the image under the scale change S4,5
  • Open your book to page 190, we are going to look at example 2
section 3 7 composition of functions
Section 3-7 Composition of Functions
  • If f and g are functions, the composite of g with f is defined by (g ◦ f)(x) = g(f(x))
  • Ex1. f(x) = 3x + 6 and g(x) = x² ─ 1, a) find f(g(5)) b) find g(f(5))
  • Ex2. f(x) = x² + 3x and g(x) = 2x + 1 find f(g(x))
  • Remember that g ◦ f ≠ f ◦ g (except at the point where they intersect, if they intersect)
  • i.e. composition of functions is NOT commutative (see example 1)

The domain of the composite is not necessarily the domain of either function individually

  • Ex3. if f(x) = x – 7 and find the domain of g(f(x))
section 3 8 inverse functions
Section 3-8 Inverse Functions
  • The inverse of a function is when you switch the location of the x and y variables
  • A graph of a function and its inverse are reflections over the line y = x
  • To see if the original graph is a function, perform a vertical line test
  • To see if the image is a function, either:
    • Perform a horizontal line test on the preimage
    • Perform a vertical line test on the image

The domain of f(x) = the range of f-1(x)

  • The range of f(x) = the domain of f-1(x)
  • Ex1. If
    • A) find the inverse
    • B) is the inverse a function?
    • C) what is the domain of the inverse
  • To test whether or not 2 functions are inverses, find f(g(x)) and g(f(x)). They must BOTH be = x
  • Ex2. Are they inverses? f(x) = 2x – 6