MCR3U 1.1 Relations and Functions Date:

Transcription

1 MCR3U 1.1 Relations and Functions Date: Relation: a relationship between sets of information. ie height and time of a ball in the air. In relations, the pairs of time and heights are "ordered"; ie ordered pairs, independent variables (x-value) and dependent variables (y- value). Relations can be represented in terms of: Equations: y = 2x - 3 Words: y is equal to two times x minus 3 Graph: Table of Values: Mapping Diagram: Function: A function is a relation in which each value of the independent variable (x-value) corresponds to exactly one value of the dependent variable (y- value). Each x-value must produce only one y-value. Domain: the set of all values of the independent variable of a relation Range: the set of all values of the dependent variable of a relation Determine which the following graphs represent a function. 1

2 Determine whether the following sets are functions Determine if the following equations represent a function Hwk: p.10 #1,2,4abc,6,7,11,12, MCR3U 1.2 Function Notation Date: To represent functions, we use notations such as f(x) and g(x). ex. Linear function: y = 2x + 1 In function notation: f(x) = 2x + 1 The notation f(x) is read f" of "x or f" at "x. The symbol f(x) represents the dependent variable (y-value). It indicates that the function f is expressed in terms of the independent variable, x. This function notation gives us more flexibility and a better way of communicating how a function needs to be evaluated ie. Old notation: y = 25x Function notation: f(x) = 25x or C(n)= 25n

3 Ex.1: Given the function (x) = x 2 4, determine: Ex.2: Consider the functions: f(x) = x 2 2x g(x) = 3x 1 b) Determine each of the following: 1.2 Assignment: P.22 #2 4a 5cd 6 7 8b(ii,iii,v,vi) 9b(iv,vi) 10 11bc c 17 3

4 MCR3U 1.3 Exploring Properties of Parent Functions Date: 4

5 5

6 6

7 1.3 Assignment: p.28 #1-3 7

8 MCR3U 1.4 Determining the Domain and Range of a Function Date: Recall: The domain of a function is the set of all first coordinates (x-values) of the relation. The range of a function is the set of all second coordinates (y-values) of the function. A quarterback throws a football from an initial height of 1.5m, the ball reaches a maximum height of 2.2m after 3.1 seconds and hits the ground at 7 seconds. State the domain and range. Ex.1: Determine the domain and range 8

9 Ex2. Given the equations of the following functions, state their domain and range. 1.4 Assignment: p.35 #1-4, 7, 11, 14a 9

10 MCR3U Sections Transformations Date: Transformations: A change made to a figure or a relation such that it is shifted or changed in shape. - Translations, reflections and stretches/compressions are types of transformations. - In order to sketch the graphs of the transformed functions, we first need to know how to graph the base function. Translations: A transformation that results in a shift (up, down, left, right) of the original figure without changing its shape. HORIZONTAL AND VERTICAL TRANSLATIONS 1) Vertical Translation of c units : * The graph of the function g(x) = f(x) + c. when c is positive, the translation is UP by c units. when c is negative, the translation is DOWN by c units. 2) Horizontal Translation of d units : * The graph of the function g(x) = f(x d). when d > 0, the translation is to the RIGHT by d units. when d < 0, the translation is to the LEFT by d units. Example 1: Given the functions f(x) graphed below, sketch the graph of g(x) on the same grid as the original function f(x). a) g(x) = f(x) - 5 b) g(x) = f(x + 3) 10

11 Example 2: Graph the following parent functions and their transformed functions. Describe the transformations that occurred. State the domain and range of the transformed functions. 11

12 REFLECTIONS OF FUNCTIONS Reflection : A transformation in which a figure is reflected over a reflection line. Invariant Points : Points that are unaltered by a transformation. 1) Reflection in the x axis or Vertical Reflection : The graph of g(x) = -f(x) Examples: Graph the following functions along with their base functions. Describe the transformations that occurred. State the domain and range. State the equations of asymptotes, if any, and state any invariant points. 12

13 2) Reflection in the y axis or Horizontal Reflection : The graph of g(x) = f(-x) Examples: Graph the following functions along with their base function. Describe the transformations that occurred. State the domain and range and equations of asymptotes, if any. 13

14 14

15 Combine Reflections and Translations Examples: For the following graphs, describe the transformations of f(x). Write the equations of the parent & transformed functions. 15

16 Examples: Graph the following functions along with their parent functions. Describe the transformations that occurred. State the domain and range of the transformed functions and the equations of the asymptotes, if any. 16

17 1.7 STRETCHES & COMPRESSIONS OF FUNCTIONS Stretches and compressions are transformations that cause functions to change shape. 1) Vertical Stretch or Compression : The graph of the function g(x) = af(x), a 0 Examples: Graph the following functions along with their parent functions. Describe the transformations that occurred. State the domain and range of the transformed functions and the equations of any asymptotes. 17

18 18

19 Examples : Graph the following functions along with their parent functions. Describe the transformations that occurred. State the domain and range of the transformed functions and the equations of any asymptotes. State any invariant points. 19

20 20

21 Mapping Rule Think about what happens to each point on the function. 21

22 1.8 COMBINING TRANSFORMATIONS OF FUNCTIONS * To perform combinations of transformations, note. Stretches, compressions and reflections can be performed in any order as long as they are performed before translations. The function must be written in the form y = af [k(x d)]+ c to identify the specific transformations. *Sometimes we need to factor out the k value to see the translation left or right. Examples: Graph the following pairs of functions using the mapping rule. Describe the transformations that occurred. 22

23 23

24 24

25 MCR3U 1.5 The Inverse Function Date: Examples : Given the following functions: a) Determine the equation of its inverse f -1 (x). b) Graph both functions and the line y = x on the same grid. c) State the domain and range of the function and its inverse. d) State whether or not the inverse is also a function. 25

26 Sometimes, the inverse of a function is not a function. For example, the inverse of a quadratic function. If we restrict the domain of the original function so that it is only half a parabola, then the inverse would be half a sideways parabola, which is a function. To restrict the domain of a quadratic function: 1) Determine the axis of symmetry. 2) Set x axis of symmetry to graph the right half of the parabola, or Set x axis of symmetry to graph the left half of the parabola. Example : Given the following functions, a) State the axis of symmetry. b) State two ways to restrict the domain so that the inverse is also a function. Example : Given the function f(x) = (x + 3) 2-2 a) Determine the inverse equation. b) Graph the original function, its inverse function, and the line y = x. c) Restrict the domain so that the inverse is also a function. d) State the domain and range of the original restricted function and its inverse function. 1.5 Assignment: p.46 #1 2ac 4bc 6cf 8 9a-e 10ef 26