TJC 2025 IP4 Math Unit 16 Graphing with GC Lesson 3 (Student)

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 1 Temasek Junior College (IP) 2025 Year 4 Mathematics Unit 16 – Graphing with Graphing Calculator (GC)1 Lesson 3: More Graphing, Graphical Solutions and Equation Solving on the GC Learning Objectives By the end of this lesson you should be able to: 1) Sketch the graphs of different functions with the aid of a graphing calculator (GC). 2) Solve equations using a GC. 3) Solve simple problems involving graphs using the GC. You may be asked to use a GC to aid you in graphing a variety of functions, both see n and unseen. As it is not possible to explore every type of function, you should learn to be competent in the use of the GC, such that you are ready to plot any function when required. In Lessons 1 and 2, we have attempted to plot some of the functions learned previously (polynomial, logarithmic, exponential, reciprocal, root etc) on the GC. In this lesson, we shall look at a few more. 3.1 Solving Inequalities using GC Sometimes we may be given questions that we are unable to solve easily using algebraic methods learnt so far. In this case, GC is able to help us visualize the problems and obtain the solutions. Example 1 (i) With the aid of a graphing calculator, sketch the graph of 32 2 5 5y x x x= + − − and 20.5 1y x x= + + , labeling coordinate of intersects and axial intercepts. (ii) Hence find the values of x that satisfies (a) 3 2 22 5 5 0.5 1x x x x x+ − − = + + , (b) 3 2 22 5 5 0.5 1x x x x x+ − −  + + . [Solution] 1 This set of notes is to be used in conjunction with the TI-84 Plus CE graphing calculator. Operation of other brands and models of graphing calculators may differ AM

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 2 (i) The coordinates of the intersection points A, B and C are ( )2.86, 2.24− , ( )0.918,0.503− and ( )2.28,5.89 respectively. (ii) (a) The values of x that satisfy 3 2 22 5 5 0.5 1x x x x x+ − − = + + are the x- coordinates of the intersection points between the 2 graphs, 2.86x=− , 0.918x=− or 2.28x= . (b) The range of values of x that satisfy 3 2 22 5 5 0.5 1x x x x x+ − −  + + are the range of values of x when the y-coordinate of the cubic graph is greater than the y-coordinate of the quadratic graph. 2.86 0.918x−  − or 2.28x . y x 1 −5 −3.09 −0.837 1.93 A B C

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 3 Exercise 1 With the aid of a graphing calculator, (a) sketch the graph of 2e1xy −=− and ( )3log 1yx=− . (b) solve ( ) 2 3e 1 log 1x x− − = − . [Solution] By zooming in closer to the region highlighted above, we see that there are two points of intersection instead of just one point of intersection. The coordinates of the intersection points between the graphs of 2e1xy −=− and 2e1xy −=− are ( )1.91, 0.0884− and ( )2,0 .  The solutions for ( ) 2 3e 1 log 1x x− − = − are 1.91x= or x = 2. Note: this question was meant to highlight the importance of checking if there is more than one intersection point, especially when the GC does not give you a clear picture in standard view. y x x = 1 y = −1 2 −0.865

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 4 Exercise 2 With the aid of a graphing calculator, solve the pair simultaneous equations ( ) 2 2 53 yx yx  = = − − [Solution] Using GC, the solutions are 2.84x= or 4x= or 5.77x= or 7.39x= . 3.2 Modulus Functions On the GC, the modulus is denoted as abs( and has the same symbolic form as the modulus notation when written. It can be accessed using ALPHA WINDOW 1 or MATH ► 1 (the 1st option in the NUM sub-menu). To make entries outside the modulus notation, press the ◄ or ► buttons until the cursor is outside the modulus notation. 22 O 3.27 6.73 y x

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 5 Exercise 3 (i) With the aid of a graphing calculator, sketch the graph of 2yx=− and 21yx=+ labeling coordinate of intersects and axial intercepts. (ii) Hence find the values of x that satisfies 2 2 1xx− = + . (iii) Using an algebraic method, find the values of x that satisfies ( ) ( ) 22 2 2 1xx− = + . What do you notice about your answers in (iii) and in (ii)? (iv) With the aid of a graphing calculator, explain if the solution to 2 2 1xx− = + satisfy the equation 2 2 1xx− = + . (v) Using your answers in (ii), find the values of x that satisfies 2 2 1xx−  + . [Solution] (i) (ii) 3x=− or 1 3x= (iii) ( ) ( ) 22 2 2 1xx− = + 22 4 4 4 4 1x x x x− + = + + 23 8 3 0xx+ − = ( )( )3 1 3 0xx− + = 1 3x= or 3x=− The solutions for part (ii) and (iii) are the same. Letting ( ) ( ) 22 2 2 1xx− = + is a possible method to solve the equation 2 2 1xx− = + . (iv) The graphs of 2yx=− and 21yx=+ only intersect at 15,33   , and not at 3x=− . Therefore, not all solutions for 2 2 1xx− = + will satisfy the equation 2 2 1xx− = + . Therefore, we must check additionally if the condition 2 1 0x+ i.e. 1 2x− is satisfied. (v) From graph, 13 3x−   . y x 2 2 1

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 6 3.3 Trigonometric Functions Recall the following: • Graphs of the sine, cosine and tangent functions are periodic. • Graphs of the sine and cosine functions have maximum and minimum points while graphs of the tangent function have asymptotes. • Graphs of sine and tangent functions exhibit rotational symmetry while graphs of the cosine function exhibit line symmetry. The sine, cosine and tangent functions can be accessed via the SIN, COS and TAN buttons. In most circumstances, you will need to graph trigonometric functions over a range of values. To ensure a good display on the GC, the following steps can be taken in sequential order: • Use ZOOM 7 , ZTrig, to obtain a better zoom of the function. • Use WINDOW to adjust Xmin and Xmax to the given range. IMPORTANT! • Always check that the GC is in the correct angle measurement (degree mode or radian mode) before graphing a trigonometric function • When required to graph BOTH trigonometric and non-trigonometric functions on the GC, ensure that your GC is SET TO RADIAN MODE. Use ZTrig for better zoom Adjust the range

2025 Year 4 Math / Graphs Unit 16 Graphing with GC Page 7 Example 5 (Self-Read) With the aid of a graphing calculator, sketch the graph of ( )3sin 2yx= for ππ x−   , stating the turning points and intersections with the axes. State (i) the maximum and minimum values of y, and (ii) the period of the function. By adding a suitable line, (iii) deduce the number of solutions to ( )3sin 2 1 2 0xx− + = for ππ x−   , and (iv) solve the equation ( )3sin 2 1 2 0xx− + = for ππ x−   , giving your answer to 3 significant figures. [Solution] • Enter the function ( )3sin 2yx= on the graphing interface. • Press ZOOM 7 to set the display to ZTrig. The graph will be displayed automatically. • Adjust the display using WINDOW to change Xmin to π− and Xmax to π . • Press GRAPH to display the graph with the adjusted settings. To obtain Xmin and Xmax above, use (-) 2ND ^ to get π− and 2ND ^ to get π . (i) To obtain the maximum and minimum values, consider ( )1 sin 2 1 x−   . ( )3 3sin 2 3 x−   Hence the maximum value of y is 3 and the minimum value of y is 3− . There is no need to use the GC to locate the maximum and minimum values in this case since the range of values of y can be worked out quickly (as long as there is a sound understanding of the sine function in this case).

2025 Year 4 Math / Graphs Unit 16 Graphing with GC