Raffles Institution WS2 Graphical Solutions of Equations

Page 1 of 21 RAFFLES INSTITUTION RAFFLES PROGRAMME 2024 YEAR 3 M ATHEM ATICS TOPIC 2: GRAPHICAL SOLUTIONS OF EQUATIONS (MATHS 1) WORKSHEET 2 Name: Class: 3 ( ) Date: WORKSHEET 2: GRAPHICAL SOLUTIONS OF EQUATIONS (NSM3 8th Edition Chapter 5, P.119) In this worksheet, you will learn how to use graphs of basic functions to solve equations. You will also learn how to use graphs of basic functions to find gradient, solve inequalities and solve further problems. Important note: Graphs will be provided and hence there is no need to use graph paper to plot graphs. (1) GRADIENT OF A CURVE (NSM3 8th Edition Chapter 5, Section 5.4, P.134) Recall: Gradient of a line gives an indication of the steepness or slope of a line. Given two points A(x1, y1) and B(x2, y2), gradient of straight line AB =21 21 yy xx

. Any two points on the same straight line will give the same gradient and hence gradient of a straight line is a constant. How about gradient of a curve? ENDURING UNDERSTANDING(S) Students will understand that xthe Cartesian coordinates system is a means of relating algebra and geometry, xalgebraic equations and graphs are different ways of representing the relationship between two or more quantities, xalgebraic equations can be solved graphically by finding the points of intersection of their graphs and xgraphs are tools which gives a visual representation of how variables are related in various real-life applications. LEARNER OUTCOMES At the end of the lesson, students will be able to xdraw proper tangents on graphs to solve given equations, xsolve equations and inequalities involving other functions such as cubic, reciprocal and exponential functions approximately by using graphical methods, xfind gradient at a point on a graph of other functions.

Page 2 of 21 Definition of Tangent to a Curve: The tangent to a curve at point P is defined as the straight line that touches the curve at P. Definition of Gradient of a Curve: Gradient of a curve at a given point is defined as the gradient of the tangent to the curve at that point. Since the tangents to the curve are different at different points, gradient of a curve changes from point to point and hence it is not a constant. Gradients obtained by drawing tangents are only approximate values. EG 1 The diagram shows the graph of y = x2 – x – 6 for – 4 d x d 4. x 2 4 x 1010

Page 3 of 21 Use the graph to answer the following questions. (a) Find the gradient of the curve at x = 2, (b) By drawing a suitable tangent, find the x-coordinate of the point on the graph at which the gradient of the tangent is equal to 1. Notes: 1. The tangent drawn should touch the curve at the given point. 2. The two points chosen to calculate the gradient should lie on the intersection of the grid lines on the graph paper. 3. As part of proper presentation for the calculation of gradient, the co ordinates of the points chosen should be clearly indicated or a triangle should be drawn with the chosen points. 4C101 2 o a Kcoordinate I

Page 4 of 21 HOMEWORK 1 LEVEL 2 1. The diagram below shows part of the graph of 2810yx x for 15xdd. Use the graph to answer the following questions. (a) Find the gradient of the curve at x = 4, (b) Find the x-coordinate of the point on the curve where the gradient is zero. [Ans: (a) –7.5 (b) x = 1.6 ] 14 l f2.1 3,0 481 Draw11 4 I 8A x 1 64

Page 5 of 21 (2) GRAPHICAL SOLUTIONS OF EQUATIONS EG 2 The diagram below shows part of the graph of 2 2412yx x for 15xdd. (a) Use the graph to solve the equation 2 2418x x . (b) By adding a suitable straight line to the graph, solve the equation xx x322 24 24 0 . X yG x n I 318 2 34izeg fromgraph Drawyo 7622k24310 Fromgraph141.2 or3.2 223 2K 24 It312 2 12 Drawy 1224

Page 6 of 21 EG 3 The diagram shows the graph of y = x2 – 2x – 4 for – 3 d x d 6. 26 x y5 a 16

Page 7 of 21 Use the graph to answer the following questions. (a) (i) Find the x-coordinates of the points on the curve where y = 5. (ii) Write down and simplify the equation in x which has these values as its solutions. (b) Find the range of values of k such that the equation x2 – 2x – 4 = k has (i) two distinct solutions, (ii) no solution. (c) Find the range of values of k such that equation 2 24 5xx x k has only one solution. (d) Find the range of values of k such that equation x2 – 2x – 4 = kx + 5 has two distinct solutions. K 2.2g K2K45 2221L904 fromgraph 5Lk 4 Fromgraph 5K204 K22K 4 5 6 drawy 5xtk fromgraph k 16or 9k24 x22K4 Kx5 25k2 Drawy Ksc5 smallestk 15s greatestk 205 GO f2 I 2.5

Page 8 of 21 HOMEWORK 2 LEVEL 2 1. The diagram below shows the graph of 2113 5 yx x for 0 14xdd. Use the graph to answer the following questions. (a) (i) Find the x-coordinates of the points on the curve where y = 4. (ii) Write down and simplify the equation in x which has these values as its solutions. y4 1.8.0 11.2.01 Drawy 4 K 1.8or K11.24 yf11311L suby 4 4 113Kx 20 132 x2 LL I3Lt20 04

Page 9 of 21 (b) Find the range of values of k such that the equation 213 10xx k has two distinct solutions. [Ans: (a)(i) 1.8 or 11.2 (ii) 2 13 20 0xx (b) 0 d k < 4.2] 2 The diagram shows part of the curve 22 12yx x for 0 3.4xdd. 132LK2 cok FCKA Fk Drawy 2k OS2k 8.4 ofkc4.2

Page 10 of 21 Use the graph to answer the following questions. (a) Find the gradient of the curve at the point where 2x by drawing a tangent. (b) By drawing a suitable straight line, find the values of x for which 212 3xx x . (c) Find the greatest possible value of y and the corresponding value of x in the range. 22 xxx X x IG 1.4 2.4512.751 DrawK 2 I 83,1 K12122 103 many 4 ok 2 y 2hr12522 2 6 9 12 72 2.454