1.1: Basic Functions and Translations

Transcription

1 .: Basic Functions and Translations Here are the Basic Functions (and their coordinates!) you need to get familiar with.. Quadratic functions (a.k.a. parabolas) y x Ex. y ( x ). Radical functions (a.k.a. square root function) y x Ex. y x Absolute-value functions y x Ex. y x 4. Reciprocal functions y Ex. x y x Note: You are expected to remember the shape of the above (left) functions!

2 Regardless of the type of function y f () x, the transformed function y f (x h) k tells us: h = k = Ex. : Find the equations for the base functions and their transformed graphs. a) Base function: Transformed function: b) Base function: Transformed function: c) Base function: Transformed function:

3 Ex. : For the function y 4 f (x ) state the value of h and k that represent the horizontal and vertical translations applied to y f (x) Ex. 3: Determine the new function when yf( x6) is translated 4 units to the left and units downward. Ex. 4: Transform the following graph. Describe the transformations in words. Given: y f () x Graph: y f ( x ) 3 Transformation: **To translate, choose key points on the graph and then translate each one to graph its corresponding image point on the transformed graph. 3

4 . (part ): Vertical and Horizontal Reflections A reflection can be identified with a negative sign. A reflection is a mirror image of a given function. Using a graphing calculator, let s explore the effect of having a negative sign at different locations of a function. Ex. : Graph y x and y x on the same grid. Ex. : Graph y x and y x on the same grid. Ex. 3: Graph y x and y x on the same grid. 4

5 Observations: y f (x) Mapping: (, ) y f (x) Mapping: (, ) Ex. 4: Without using a graphing calculator, a) graph f ( x) x3 with a solid line. b) graph y f () x with a dotted line.... c) graph y f ( x) with a broken line. _ f x 4x x, write a new function after applying the following reflection: Ex. 5: Given 3 a) over the x-axis b) over the y-axis 5

6 Ex. 6: Given f (x), graph the indicated relation. State the domain and range for each of them. Determine if it is a function. a) Graph y f x Domain: Range: Function? b) Graph y f x Domain: Range: Function? Homework: p. 8 9: #, 3, 4, 5c, 5d, 7b, 7d * use graph paper when drawing graphs (available from Ms. Dobson) * mapping notation : (x,y) (, ) 6

7 . (part ): Expansions and Compressions Ex. : Consider the following function: f x x Graph the indicated function using the table of values on the graphing calculator: a) y f x b) y f x c) y f x d) y f x 7

8 Observations: y af (x) Mapping: (, ) y f (bx) Mapping: (, ) Ex. : Using y f (x) as a base function, describe the transformation when x is replaced by x and y is replaced by 3 y in words and in mapping notation. Ex. 3: Given the graph of f (x), perform each of the following transformations: a) a vertical expansion by a factor of b) a horizontal compression by a factor of Homework: p. 8 3: #, 5a, 5b, 6, 7a, 7c, 8, 9, 4 * use graph paper when drawing graphs (available from Ms. Dobson) 8

9 .3 (part ): Combining Transformations Does order matter? Let s explore. Ex. : Graph y x x y Vertical Expansion first vs. Vertical Translation first: (Reverse Order) ) VE by factor of ) VT by unit down ) VT by unit down ) VE by a factor of Did the order effect the outcome? Horizontal Expansion first vs. Horizontal Translation first: (Reverse Order) ) HE by factor of ) HT by unit right ) HT by unit right ) HE by a factor of Did the order effect the outcome? 9

10 Build-it-up Method: Replace x with new x Put in new y setting. Ex. : Given the description, write the following transformations in function notation. a) VE of b) HE of VT down 3 VC of /6 HT right VT up HC of /3 HT left 6 Combinations of Transformations: We will perform transformations in the order S (Stretches), R (Reflections), T (Transformations) First, re-write the function as y af (b(x h)) k to be able to read all of the transformations directly. Notice: there is no coefficient on x it must be factored out! We will perform the transformations in order SRT for both vertical and horizontal: ) A Vertical Stretch by a factor of a; a Horizontal Stretch by a factor of ) A Vertical Reflection if a < 0 in the x-axis; a Horizontal Reflection if b < 0 in the y-axis 3) A Vertical Translation by a factor of k; a Horizontal Translation by a factor of h b Ex. 3: Describe the order of transformations that occur for the following functions. a) y 3f x 4 b) y f x 5 3 c) y f x d) y f x 0

11 Ex. 4: Given y f ( x) : a) Describe the transformation y f x 3 b) Graph the indicated functions on the grid provided. Ex. 5: Write the equation for both the base function and the transformed function.

12 Math Pre-Calculus.4: Inverse Functions The inverse of a relation is found by interchanging the x and y coordinates of the ordered pairs of that relation. Recall: an invariant point is Mapping: (x,y) (, ) any point that is unchanged Graphically, this is the same as a reflection across the line y = x The notation for an inverse is f ( x) Ex. : Given the function f ( x) x 3: a) Determine f ( x) b) Graph f (x) and f ( x ) on the same grid c) Invariant point(s): Ex. : Consider the function f ( x) x 4 a) Graph f(x) on the grid provided and state the domain and range. b) Find the inverse of the function f ( x) x 4. c) Graph the inverse and state the domain and range. d) Is the inverse of f a function? If not, how could the domain or range of f(x) be restricted so that the inverse of f is a function?

13 Math Pre-Calculus Ex. 3: For each of the following functions Find the inverse function using the notation f ( x ), where appropriate State the domain and range of the given function and its inverse. a) f ( x) x b) f ( x) ( x 3) c) f( x) x 7 x 3 Ex. 4: a) Accurately graph the inverses of the following functions on the same grid. b) Is the inverse a function? Justify your answer Homework: p. 5 53: #, 4ac, 5ace, 7a, 9ce,, 3 (use graphing calculator), 5 3