FUNCTIONS Concepts in Functions Straight Line Graphs Parabolas Hyperbolas Exponentials Sine Graphs Cos Graphs Tan Graphs

CONCEPTS IN FUNCTIONS Formulae and Tables x -2 1 2 3 y -4 4 6 E.g. Given the formula y = 2x, the table can be constructed: x -2 1 2 3 y -4 4 6 x in this relation is called the independent variable, since the values of x were chosen randomly. However, it is clear that the values of y depended entirely on the values of x as well as the rule used, namely, y = 2x. In this relation, y is called the dependent variable.

Ordered Pairs on the Cartesian Plane (Co-ordinates) Graphs In order to draw graphs, we need to plot co-ordinates or ordered pairs onto the Cartesian Plane Co-ordinate or Ordered Pair: (independent variable; dependent variable) (x; y) Ordered Pairs on the Cartesian Plane (Co-ordinates)

Plotting points on a Cartesian Plane The Cartesian Plane Decode the message by writing down the letter found at each of the following coordinates: (1,5), (3,1), (3,4), (5,2), (5,4), (1,1), (4,5) and (1,2). Plotting points on a Cartesian Plane

Domain and Range Domain refers to the x-values that exist Range refers to the y-values that exist Examples a) Given: set of ordered pairs {(-2; 16); (0; 4); (3; 7)} Domain = {-2; 0; 3} Range = {16; 4; 7} b) Given: equation Domain = {-2; -1; 0; 1; 2; 3} Range = {-4;-2;0;2;4;6}.

Finding the Domain and Range of Graphs b) Given: graph Domain: Range: Finding the Domain and Range of Graphs

f(x) Notation f(x) is the equivalent of y Example: Given determine the value of: a) f(2) … therefore, substitute x = 2 b) f(x)=0

EXERCISE Determine the domain and range: a) Given the set: {(-3;-4); (2;1); (-6;-9)} b) Given the equation: y=4x-6 c) Given the graph:

2. If determine the value of: a) f(1) b) f(- 1) c) f (2x) d) f(x)=0 e) f(x)=1

STRAIGHT LINE GRAPHS The graph of a linear function is a straight line. Standard form of a straight line graph: y = m x + c where m = gradient and c = y-intercept

Effects of m and c in y = mx + c: m represents the gradient of the straight line If m > 0, the gradient is positive If m < 0, the gradient is negative c represents the y-intercept of the straight line graph and also indicates the vertical translation of the graph If c > 0, the graph has a positive y-intercept and shifts up by c units If c < 0, the graph has a negative y-intercept and shifts down by c units Investigating the effects of m in a Straight Line Investigating the effects of c in a Straight Line

Sketching Straight Line Graphs Using The Table Method Example 1 Sketch the graph of y = 2x+4 by using the table method. x -1 1 2 y 4 6 8 The x-values were randomly chosen. The y-values were found by substituting the x - values into the equation y = 2x +4.

Note! x-intercept is at (-2; 0) y-intercept is at (0; 4)

Sketching Straight Lines using the Dual-Intercept Method Sketching Straight Line Graphs Using The Dual-Intercept Method Sketching Straight Lines using the Dual-Intercept Method Example Sketch the graph of y = -3x - 3 by using the dual- intercept method. x-intercept: y = 0 y-intercept: x = 0 0 = -3x – 3 y = -3(0) – 3 3x = -3 y = 0 – 3 x = -1 y = -3 (-1; 0) (0;-3)

The Graph of y = -3x – 3:

Sketching Straight Line Graphs Using The Gradient-Intercept Method Example Sketch the graph of by using the gradient- intercept method. The negative sign indicates that the line slopes to the left. The y-intercept is 0. The numerator tells us to rise up 1 unit from the y - intercept. The denominator tells us to run 2 units to the left Determining the Gradient using the Gradient-Intercept Method

Horizontal and Vertical Lines a) Sketch the graph of y = 2 In this relation, the x-values can vary but the y - values must always remain 2. Lines which cut the y - axis and are parallel to the x – axis have equation y = number x -1 1 2 y Determining the Gradient of Horizontal Straight Lines Horizontal Straight Lines

Vertical Straight Lines b) Sketch the graph of x = 2 In this relation, the y-values can vary but the x - values must always remain 2. Lines which cut the x - axis and are parallel to the y – axis have equation x = number x 2 y -2 -1 1 Vertical Straight Lines

Straight Line Graphs Calculator EXERCISE 1. Draw neat sketch graphs of the following linear functions on separate axes. (a) f (x) = 3x- 6 (f) y - 3x = 6 (b) g (x) = - 2x + 2 (g) y = 3x + 2 (c) h (x) = - 4x (h) 2x + 3y + 6 = 0 (d) 5x + 2y = 10 (i) 3x + 3y = 0 (e) y – x = 0 (j) 3x = 2y Straight Line Graphs Calculator

With each of the equations below, do the following: Write the equation in standard form Determine the gradient Determine the y-intercept (a) 2y - 4x = 0 (c) 6x - 3y = 1 (b) 2y + 4x = 2 (d) x - 2y = 4 Given the following equation: f (x) = - 2x + 1 (a) Sketch the graph (b) Draw the graph of g(x) if it is f(x) the has been translated 2 units upwards.

Finding the Equations of Straight Line Graphs Determining the Equation of a Straight Line Graph Example 1: Determine the equation of the line: The y-intercept is 3, so c = 3. y = m x + 3 Substitute the point (8 ; - 1) - l = m(8) + 3 - 1 = 8m + 3 - 8m = 4 m = -½ Therefore the equation is : y = - ½ x +3 Calculating the Gradients of Different Staircases Challenge!Finding the gradient when c = 0

EXERCISE (a) (b) (c ) (d) 1. Determine the equations of the following lines: (a) (b) (c ) (d)

2. Determine the equation of the line: a) passing through the point (-1; - 2) and cutting the y-axis at 1. b) Determine the equation of the line with a gradient of - 2 and passing through the point (2; 3). c) Determine the equation of the line which cuts the x-axis at 5 and the y - axis at - 5. d) Determine the equation of the line which cuts the x-axis at – 3 and the y - axis at 9.

Parallel Lines Parallel lines have the same gradient (m) 1. Are these lines parallel? y = 2x + 4 and y – 2x – 6 = 0 y = 2x + 4 => m = 2 y – 2x – 6 = 0 y = 2x + 6 => m = 2 Yes! They are parallel. 2. Are these lines parallel? 2y = 2x + 4 and – 2x + y – 6 = 0 2y = 2x + 4 y = x + 2 => m = 1 - 2x + y – 6 = 0 y = 2x + 6 => m = 2 No! They are not parallel. Parallel lines

Perpendicular Lines   Perpendicular Lines

Intersecting Straight Line Graphs Point of Intersection: The point at which both straight line graphs have the same value for x and for y Can determine the point of intersection a) graphically b) algebraically (solve equations simultaneously) Example: Solve 3x-y=4 and 2x-y=5 a) Graphically Point of intersection

Points of Intersection Calculator Example: Solve 3x-y=4 and 2x-y-5 b) Algebraically 3x – y = 4 … (1) -y = 4 – 3x y = 3x – 4 … (3) 2x-y=5 … (2) Substitute (3) into (2) 2x – (3x-4) = 5 2x – 3x + 4 = 5 -x = 1 x = -1 Substitute x = -1 into (3) y=3(-1)-4 y =-7 Point of intersection: (-1;-7) Points of Intersection Calculator

EXERCISE 1. Draw neat sketch graphs of the following lines on the same set of axes: x + y = 3 and x - y = -1. 2. Solve x + y = 3 and x - y = - 1 using the method of simultaneous equations. 3. Determine the coordinates of the point of intersection of the following pairs of lines: x + 2y = 5 and x- y = - 1

Parabolas Quadratic Equations have the formula: y = ax² + q Investigation 1 Complete the following table and then draw the graphs of each function on the set of axes provided below. x -2 -1 1 2 x² 2x² 3x²

Investigating the effects of a of a Parabola x² 2x² 3x² Investigating the effects of a of a Parabola

Investigation 2 Complete the following table and then draw the graphs of each function on the set of axes provided below. x -2 -1 1 2 x² - 2 x² x² + 2

Investigating the effects of q of a Parabola x² x²+2 x²-2 Investigating the effects of q of a Parabola

Example Given: (a) Write down the coordinates of the y - intercept. (b) Determine the coordinates of the x - intercepts. 0 = (x + 2)(x - 2) x + 2 = 0 or x – 2 =0 x= - 2 or x = 2 x-intercepts : (- 2; 0) and (2; 0) (c) Sketch the graph.

Given: (d) Determine the turning point of the graph. Turning point is (0; - 4) (e) Determine the minimum value of the graph. Minimum value = -4. (f) Determine the domain of the graph. (g) Determine the range of the graph.

Reflecting Straight Lines & Parabolas   Reflecting Straight Lines & Parabolas

THE HYPERBOLA Standard form of a hyperbola graph: x -6 -4 -2 2 4 6 y Investigation 1: e.g. Complete the table using substitution: x -6 -4 -2 2 4 6 y - 1/3 - ½ -1 undefined 1 ½ 1/3

x -6 -4 -2 2 4 6 y Investigation 2: e.g. Complete the table using substitution: x -6 -4 -2 2 4 6 y 2,67 2,5 2 undefined 4 3,5 3,33

Hyperbolas

EXERCISE Sketch the following, using the table method: 1) 2)

Standard form of a hyperbola graph: a > 0: Quadrants 1 & 3 a < 0: Quadrants 2 & 4 q > 0: Graph shifted up q < 0: Graph shifted down NOTE! Asymptote on y-axis (x = 0)

EXPONENTIALS Standard form of an exponential graph: y = a x + q x -3 Investigation 1: e.g. y = 3 x Complete the table using substitution: x -3 -2 -1 1 2 3 y 0,037 0,111 0,333 1 3 9 27

y = 3x

y = 0 is the equation of the asymptote y = 3x Write down the co-ordinates of the x intercept. X intercept : This graph does not cut the x – axis. y = 30 = 1 ie y 0 y = 0 is the equation of the asymptote

y = 3x y = 0

Sketch the graph of: 1) y = 2 -x 2 ) y = ½ x What do you notice? It’s the same graph! Why? Because of our Exponent Laws!

Sketch the graph of y = 3x - 1 What do you think it will look like ? How will it differ from y = 3x ? What is the equation of its asymptote ?

Investigating the Exponential Graph y = 3x - 1 y = -1 Investigating the Exponential Graph

Standard form of an exponential graph: y = a x + q a is a whole number: e.g. y = 2 x (increasing function) a is a fraction: e.g. y = ½ x or y = 2 -x (increasing function) q > 0: Graph shifted up q < 0: Graph shifted down NOTE! Asymptote on x-axis (y = 0)

Example 3 PLEASE ANIMATE Find the value of a and q in the equation of the exponential graph: y = a.2x+1+q q = -3 Subst. (-1;0) 0 = a.2-1+1- 3 0 = a.1 – 3 3 = a

Example 2 PLEASE ANIMATE Find the value of b and q in the equation of the exponential graph: y = -3.bx+1+q q = 1 Subst. (-2;0) 0 = -3.b-2+1+ 1 0 = -3.b-1 + 1

SINE GRAPHS The trig function “Sine” can be sketched Example 1: a) Complete the table of y = sinx for x [0°; 360°] 0° 30° 45° 60° 90° 120° 135° 150° 180° y = sin x 210° 225° 240° 270° 300° 315° 330° 360° y = sin x

Link between sinx as a ratio and as a graph

b) Determine the following based on the graph: The maximum value is…. 1 The minimum value is … -1 Amplitude = (maximum - minimum value) divided by 2 = (1-(-1)) 2 = 2 2 =1 c) Determine: The domain of the graph is … The range of the graph is …

Example 2: a) Draw the graph of y = sin x for x [-360°; 360°] b) After how many degrees does the graph (pattern) start repeating itself? … 360° This is called the period of the graph.

Exercise Graph Amplitude Maximum Minimum Period y = sin x y = 2 sin x y =3 sin x y = - sin x y = - 2 sin x The effects of a in y = asinx Amplitude shifts of y = asinx

Vertical shifts of y = sinx + q Exercise Graph Amplitude Maximum Minimum Period y = sin x y = sin x + 1 y = sin x - 2 Vertical shifts of y = sinx + q

COS GRAPHS The trig function “Cosine” can be sketched Example 1: a) Complete the table of y = cosx for x [0°; 360°] 0° 30° 45° 60° 90° 120° 135° 150° 180° y = cos x 210° 225° 240° 270° 300° 315° 330° 360° y = cos x

Link between cosx as a ratio and as a graph

b) Determine the following based on the graph: The maximum value is…. 1 The minimum value is … -1 Amplitude = (maximum - minimum value) divided by 2 = (1-(-1)) 2 = 2 2 =1 c) Determine: The domain of the graph is … The range of the graph is …

Example a) Draw the graph of y = cosx for x [-360°; 360°] b) After how many degrees does the graph (pattern) start repeating itself? … 360° This is called the period of the graph.

Exercise Graph Amplitude Maximum Minimum Period y = cos x y= 2 cos x y = - 2 scos x

Exercise Graph Amplitude Maximum Minimum Period y = cos x

TAN GRAPHS The trig function “Tangent” can be sketched Example 1: a) Complete the table of y = tan x for x [0°; 360°] x is undefined at 90◦ and 270◦ Therefore, the tan graph has ASYMPTOTES at x = 90◦ and x = 270◦ 0° 45° 90° 135° 180° y = tan x 225° 270° 315° 360° y = tan x

b) Determine the following based on the graph: The maximum value is…. There is none! The minimum value is … There is none! Amplitude - There is none! However: y = atanx … a is the “amplitude” c) Determine: The domain of the graph is … The range of the graph is …

Example a) Draw the graph of y = tan x for x [-360°; 360°] b) After how many degrees does the graph (pattern) start repeating itself? … 180° This is called the period of the graph.

Exercise Graph Amplitude Maximum Minimum Period y = tan x y= 2 tan x

Exercise Graph Amplitude Maximum Minimum Period y = tan x