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Sketching Graphs in C1 - IGCSE FM Lesson Overview by Dr. J. Frost

Learn specific skills in sketching quadratics, cubics, and reciprocals. Identify features, roots, y-intercepts, max/min points, and asymptotes. Explore graph transformations and piecewise functions. Practice with examples and exercises.

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Sketching Graphs in C1 - IGCSE FM Lesson Overview by Dr. J. Frost

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  1. IGCSE FM/C1 Sketching Graphs Dr J Frost (jfrost@tiffin.kingston.sch.uk) Objectives: (from the IGCSE FM specification) Last modified: 27th August 2015

  2. Overview Over the next 5 lessons: #2: Specific skills in sketching (i) quadratics (ii) cubics and (iii) reciprocals #1: Shapes of graphs (quadratic, cubic, reciprocal) and basic features (roots, y-intercept, max/min points, asymptotes) C1 IGCSE FM C1 IGCSE FM #4: Graph transformations C1 only #3: Piecewise functions IGCSE FM only

  3. #1 :: Features of graphs There are many features of a graph that we might want to identify when sketching. y-intercept? ? y as ? ? y as ? ? Turning Points? ? Roots? ? Asymptotes? ? ! An asymptote is a straight line that a curve approaches at infinity (indicated by dotted line).

  4. #1 :: Types of graphs There are three types of graphs you need to be able to deal with in C1 and/or IGCSE FM: e.g. e.g. e.g. Parabola (Quadratic Equation) Cubic Reciprocal At GCSE these were previously centred at the origin.

  5. RECAP :: Sketching Quadratics y 3 features needed in sketch? Roots ? x ? General shape: Smiley face or hill? y-intercept ?

  6. Example 1 Roots y-intercept Shape: smiley face or hill? ? y x -1 2 So if , i.e. , then or . -2 When , clearly .

  7. Example 2 Roots y-intercept Shape: smiley face or hill? y ? = ? x 1 4 Bro Tip: We can tidy up by using the minus on the front to swap the order in one of the negations. -4

  8. Test Your Understanding So Far ? Roots? x = -1, -2 ? Roots? ? ? y = 2 y-Intercept? y-Intercept? ? ? or shape? or shape? ? y y ? 8 2 x x -2 -1 -2 4

  9. Graph Equation Find an equation for this curve, in the form where are integers. Find an equation for this curve, in the form where are integers. ? ?

  10. Understanding features of a quadratic IGCSE FM June 2012 Paper 2 Q4 ? Positive (to give shape) ? Negative (this is -intercept) Since the solutions are the roots. Thus (using the graph) one is positive, one negative. ? From graph, never drops below -3, so 0 solutions. ? ? Line is horizontal. So tangent is

  11. RECAP :: Using completed square for min/max ? How could we use this completed square to find the minimum point of the graph? (Hint: how do you make as small as possible in this equation?) ? ? Anything squared must be at least 0. So to make the RHS as small as possible, we want to be 0. This happens when . When , .

  12. Write down ! When we have a quadratic in the form: The minimum point is .

  13. Complete the table, and hence sketch the graphs Equation Completed Square x at graph min y at graph min y-intercept Roots? 1 y = x2 + 2x + 5 y = (x + 1)2 + 4 -1 4 5 None 2 y = x2 – 4x + 7 ? y = (x – 2)2 + 3 ? 2 ? 3 ? 7 None ? ? ? ? ? ? 3 y = x2 + 6x – 27 y = (x + 3)2 – 36 -3 -36 -27 x = 3 or -9 1 2 ? 3 ? 7 3 -9 5 (2,3) (-1,4) -27 (-3,-36)

  14. Exercise 1 (Exercises on provided sheet) Sketch the following parabolas, ensuring you indicate any intersections with the coordinate axes. If the graph has no roots, indicate the minimum/maximum point.(a) (b) (c) (d) (e) 1

  15. Exercise 1 (Exercises on provided sheet) Find equations for the following graphs, giving your answer in the form Sketch the following parabolas. These have no roots, so complete the square to identify the minimum/maximum point.(a) (b) 2 3 ? a ? b ? ? c ?

  16. Exercise 1 (Exercises on provided sheet) [C1 May 2010 Q4] (a) Show that x2 + 6x + 11 can be written as (x + p)2 + q, where p and q are integers to be found. (2) (b) Sketch the curve with equation y = x2+ 6x + 11, showing clearly any intersections with the coordinate axes. (2) (c) Find the value of the discriminant of x2 + 6x + 11. (2) [AQA] The diagram shows a quadratic graph that intersects the -axis when and .Work out the equation of the quadratic graph, giving your answer in the form where are integers. 4 5 ? ? ? ?

  17. Exercise 1 (Exercises on provided sheet) A parabola has a maximum point of .(a) Given the quadratic equation is of the form , determine and . (b) Determine the discriminant. 6 [Set 2 Paper 2] Here is a sketch of (a) Write down the two solutions of (b) Write down the equation of the line of symmetry of 7 ? ? ? ?

  18. #2b :: Sketching Cubics A recap of their general shape from GCSE… When When y ? ? x When When y ? ? x

  19. #2b :: Sketching Cubics • Is it uphill or downhill? Is the term + or -? • Consider the roots: • If appears once, the line crosses at . • If appears, the line touches at . • If appears, we have a point of inflection at . y y ? ? 2 x x -1 1 -2 1

  20. More Examples • Is it uphill or downhill? Is the term + or -? • Consider the roots: • If appears once, the line crosses at . • If appears, the line touches at . • If appears, we have a point of inflection at . y ? ? y x x 1 -1 2 -1 A point of inflection is where the curve changes from concave to convex (or vice versa). Think of it as a ‘plateau’ when ascending or descending a hill.

  21. Test Your Understanding Sketch , ensuring you indicate where the graph cuts/touches either axes. Suggest an equation for this graph. ? -3 ? 4 Sketch (hint: factorise first!) ? -1 2 -3 3

  22. Quickfire Questions! Sketch the following, ensuring you indicate the values where the line intercepts the axes. 1 4 6 ? 27 ? ? 6 3 -2 1 3 1 2 7 5 ? ? ? 1 -2 1 2 -4 8 3 ? ? 3 -1 1 3

  23. Exercise 2 (Exercises on provided sheet) [Set 1 Paper 2] Sketch the curve 2 a 1 ? ? 0.5 3 b ? -1 12 c ? 8 -2 d ? 18 -3 2

  24. Exercise 2 (Exercises on provided sheet) [Set 4 Paper 2] A sketch of , where is a cubic function, is shown. [Set 2 Paper 2] Here is a sketch of where are constants. 4 3 There is a maximum point at .(a) Write down the equation of the tangent to the curve at .(b) Write down the equation of the normal to the curve at . Work out the values of . ? ? ?

  25. Exercise 2 (Exercises on provided sheet) 5 ?

  26. Exercise 2 (Exercises on provided sheet) 6 Suggest equations for the following cubic graphs. (You need not expand out any brackets) a b 3 -4 -2 ? ? c d -1 -3 ? ?

  27. Exercise 2 (Exercises on provided sheet) 7 ?

  28. #2c :: Reciprocal Graphs At GCSE, you encountered ‘reciprocal graphs’, with equations of the form: where is a constant. We’ll be able to sketch more complicated graphs of this form:

  29. Example Sketch Is there a value of for which is not defined? We can’t divide by 0. This occurs when . We draw a dotted line (known as an asymptote) and MUST give its equation. ? The rest of the curve will be the same as before (consider for example what happens when or ). YOU MUST WORK OUT THE INTERCEPTS.

  30. Example ? Sketch ? Increasing the values by 2 shifts the graph up. We now have a horizontal asymptote! Note we now also have a root, which we work out in the usual way by solving Sketch

  31. Test Your Understanding Sketch Sketch ? ?

  32. Exercise 3 Sketch the following, ensuring you indicate the equation of any asymptotes and the coordinates of any points where the graph crosses the axes.(a) (b) (c) (d) (e) 1 ? ? ? ? ?

  33. Exercise 3 2 ?

  34. Exercise 3 3 ?

  35. Exercise 3 4 ?

  36. #3 :: Piecewise Functions Sometimes functions are defined in ‘pieces’, with a different function for different ranges of values. Sketch > Sketch > Sketch > (2, 9) (0, 5) (-1, 0) (5, 0)

  37. Test Your Understanding Sketch Sketch Sketch This example was used on the specification itself! (2, 1) (1, 1) (3, 1)

  38. Exercise 4 (Exercises on provided sheet) • [June 2013 Paper 2] A function is defined as: • Draw the graph of for • [Jan 2013 Paper 2] A function is defined as: • Draw the graph of for • Use your graph to write down how many solutions there are to 3 sols • Solve 2 1 b ? c ? ? a ?

  39. Exercise 4 (Exercises on provided sheet) • [Specimen 1 Q4] A function is defined as: • Calculate the area enclosed by the graph of and the axis. • [Set 1 Paper 1] A function is defined as: • Draw the graph of for . 4 3 ? Sketch ? Area = ?

  40. Exercise 4 (Exercises on provided sheet) • [AQA Worksheet Q10] • [AQA Worksheet Q9] • Draw the graph of from . 6 5 ? 2 -1 -2 -3 -4 1 2 3 4 5 3 7 -1 Show that Area of Area of ?

  41. #4 :: Graph Transformations – GCSE Recap Suppose we sketch the function y = f(x). What happens when we sketch each of the following?  3 ?  2 ?  Stretch x by factor of ½ ? ↔ Stretch x by factor of 3 ? ↑4 ? ↕Stretch y by factor of 3. ? If inside f(..), affects x-axis, change is opposite. If outside f(..), affects y-axis, change is as expected.

  42. RECAP :: vs We don’t have to reason about these any differently! y = f(x) y Bro Tip: Ensure you also reflect any min/max points, intercepts and asymptotes. (2, 3) 1 x y = -1 y y ? ? Change inside f brackets, so times values by -1 (-2, 3) y = 1 x 1 -1 Change outside f brackets, so times y values by -1 x (2, -3) y = -1

  43. Test Your Understanding C1 Jan 2009 Q5 Figure 1 shows a sketch of the curve C with equation y = f(x). There is a maximum at (0, 0), a minimum at (2, –1) and C passes through (3, 0). On separate diagrams, sketch the curve with equation (a) y = f(x + 3),(3) (b) y = f(–x). (3) On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the x-axis. b ? a ?

  44. Drawing transformed graphs Bro Tip: To sketch many functions, it’s best to start with a similar simpler function (in this case ), then consider how it’s been transformed. Sketch ? y 7 x -1 If gives you a translation right by 1 unit and 8 down.

  45. Drawing transformed graphs Sketch (Hint: If , then what is the above function?) y ? x -2 -0.5 So translation 2 left 1 down.

  46. Test Your Understanding C1 June 2009 Q10 a ? a) b ? b) c) If Then This is a translation right of 2. c ? 3 2 5

  47. Exercise 5 (Exercises on provided sheet) [C1 Jan 2011 Q5] Figure 1 shows a sketch of the curve with equation where The curve passes through the origin and has two asymptotes, with equations y = 1 and x = 2, as shown in Figure 1. (a) Sketch the curve with equation y = f(x − 1) and state the equations of the asymptotes of this curve. (3) (b) Find the coordinates of the points where the curve with equation y = f(x − 1) crosses the coordinate axes. (4) 1 ? ?

  48. Exercise 5 (Exercises on provided sheet) 2 ? [C1 May 2010 Q6] Figure 1 shows a sketch of the curve with equation y = f(x). The curve has a maximum point A at (–2, 3) and a minimum point B at (3, – 5). On separate diagrams sketch the curve with equation (a) y = f (x + 3), (3) (b) y = 2f(x). (3) On each diagram show clearly the coordinates of the maximum and minimum points. The graph of y = f(x) + a has a minimum at (3, 0), where a is a constant. (c) Write down the value of a.(1) ? ?

  49. Exercise 5 (Exercises on provided sheet) 3 ? [C1 May 2011 Q8] Figure 1 shows a sketch of the curve C with equation y = f(x). The curve C passes through the origin and through (6, 0). The curve C has a minimum at the point (3, –1). On separate diagrams, sketch the curve with equation (a) y= f(2x), (3) (b) y= −f(x), (3) (c) y= f(x + p), where 0 < p < 3. (4) On each diagram show the coordinates of any points where the curve intersects the x-axis and of any minimum or maximum points. ? ?

  50. Exercise 5 (Exercises on provided sheet) 4 ? • [C1 May 2012 Q10] Figure 1 shows a sketch of the curve C with equation y = f(x), where • f(x) = x2(9 – 2x) • There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A. • (a) Write down the coordinates of the point A. • (b) On separate diagrams sketch the curve with equation • (i) y = f(x + 3), (ii) y = f(3x). • On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. • The curve with equation y = f(x) + k, where k is a constant, has a maximum point at (3, 10). • (c) Write down the value of k. ? ? ?

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