Chapter 3

1. Chapter 3 Transformations of Graphs and Data

2. 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)

3. 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.

4. 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)

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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)

13. 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

14. 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)

15. 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))

16. 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

17. 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