IGCSE FMC 1 Sketching Graphs Dr J Frost
- Slides: 52
IGCSE FM/C 1 Sketching Graphs Dr J Frost (jfrost@tiffin. kingston. sch. uk) Objectives: (from the IGCSE FM specification) Last modified: 27 th August 2015
Overview Over the next 5 lessons: #1: Shapes of graphs (quadratic, cubic, reciprocal) and basic features (roots, yintercept, max/min points, asymptotes) C 1 #2: Specific skills in sketching (i) quadratics (ii) cubics and (iii) reciprocals C 1 IGCSE FM #4: Graph transformations #3: Piecewise functions IGCSE FM only C 1 only
#1 : : Features of graphs There are many features of a graph that we might want to identify when sketching. y-intercept? ? ? ? Turning Points? ? Roots? ? Asymptotes? ? ! An asymptote is a straight line that a curve approaches at infinity (indicated by dotted line).
#1 : : Types of graphs There are three types of graphs you need to be able to deal with in C 1 and/or IGCSE FM: Parabola (Quadratic Equation) Cubic Reciprocal At GCSE these were previously centred at the origin.
RECAP : : Sketching Quadratics 3 features needed in sketch? y Roots ? x General shape: ? Smiley face or hill? y-intercept ?
Example 1 1. Roots 2. y-intercept 3. Shape: smiley face or hill? y ? -1 2 -2 x
Example 2 1. Roots 2. y-intercept 3. Shape: smiley face or hill? y ? ? Bro Tip: We can tidy up by using the minus on the front to swap the order in one of the negations. 1 4 -4 x
Test Your Understanding So Far Roots? x = -1, -2 ? Roots? y-Intercept? y = 2 ? y-Intercept? ? y ? ? ? y 8 2 -2 -1 ? x ? -2 4 x
Understanding features of a quadratic IGCSE FM June 2012 Paper 2 Q 4 ? ? ?
RECAP : : Using completed square for min/max ? ? ?
Write down !
Complete the table, and hence sketch the graphs Equation Completed Square x at graph y-intercept Roots? min 1 y = x 2 + 2 x + 5 y = (x + 1)2 + 4 -1 2 y = x 2 – 4 x + 7 y = (x – 2) ? 2 + 3 2 3 y = x 2 + 6 x – 27 y = (x + 3) ? 2 – 36 -3 1 ? ? 4 5 ? -36 ? 7 3 2 -27 None ? ? None? x = 3 or -9 ? 3 7 5 (-1, 4) ? -9 (2, 3) ? -27 (-3, -36) 3
Exercise 1 1 (Exercises on provided sheet)
Exercise 1 2 (Exercises on provided sheet) 3 a ? ? b ? c ? ?
Exercise 1 (Exercises on provided sheet) 5 4 ? ?
Exercise 1 (Exercises on provided sheet) 6 7 ? ?
#2 b : : Sketching Cubics A recap of their general shape from GCSE… y ? ? x y ? x ?
#2 b : : Sketching Cubics y y 2 ? -1 1 x ? -2 1 x
More Examples y y ? -1 2 x 1 -1 ? x 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.
Test Your Understanding ? -3 4 ? -1 2 ? -3 3
Quickfire Questions! Sketch the following, ensuring you indicate the values where the line intercepts the axes. 1 4 6 27 ? 6 -2 2 ? 1 5 ? 3 ? 3 1 2 3 1 7 ? ? ? 1 -2 -4 8 -1 3 ? 1 3
Exercise 2 (Exercises on provided sheet) 1 2 a ? 0. 5 b ? -1 12 c ? 8 ? -2 d 18 -3 ? 2 3
Exercise 2 (Exercises on provided sheet) 3 4 ? ? ?
Exercise 2 (Exercises on provided sheet) 5 ?
Exercise 2 6 (Exercises on provided sheet) Suggest equations for the following cubic graphs. (You need not expand out any brackets) a b -4 ? c 3 -2 d -1 -3 ? ? ?
Exercise 2 (Exercises on provided sheet) 7 ?
#2 c : : Reciprocal Graphs We’ll be able to sketch more complicated graphs of this form:
Example ?
Example ?
Test Your Understanding ?
Exercise 3 1 ? ?
Exercise 3 2 ?
Exercise 3 3 ?
Exercise 3 4 ?
#3 : : Piecewise Functions Sketch > (2, 9) (0, 5) (-1, 0) (5, 0)
Test Your Understanding Sketch This example was used on the specification itself! (1, 1) (2, 1) (3, 1)
Exercise 4 (Exercises on provided sheet) 2 1 c ? b ? ? a ?
Exercise 4 (Exercises on provided sheet) 4 3 Sketch ? ? ?
Exercise 4 (Exercises on provided sheet) 6 5 2 1 2 3 4 5 -1 -2 -1 ? 3 7 -3 -4 ?
#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.
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 (-2, 3) 1 y = 1 ? -1 ? x y = -1 (2, -3) x Change outside f brackets, so times y values by -1
Test Your Understanding C 1 Jan 2009 Q 5 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. a ? b ?
Drawing transformed graphs y 7 -1 ? x
Drawing transformed graphs y -2 -0. 5 ? x
Test Your Understanding C 1 June 2009 Q 10 b) a ? b ? 3 c ? 2 5
Exercise 5 (Exercises on provided sheet) 1 ? ?
Exercise 5 (Exercises on provided sheet) 2 [C 1 May 2010 Q 6] 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 = 2 f(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) ? ? ?
Exercise 5 (Exercises on provided sheet) 3 ? [C 1 May 2011 Q 8] 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(2 x), (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. ? ?
Exercise 5 (Exercises on provided sheet) 4 ? [C 1 May 2012 Q 10] Figure 1 shows a sketch of the curve C with equation y = f(x), where f(x) = x 2(9 – 2 x) 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(3 x). 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. ? ?
Exercise 5 (Exercises on provided sheet) 5 ?
Exercise 5 (Exercises on provided sheet) 6 ? [C 1 June 2008 Q 3] Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the point (0, 7) and has a minimum point at (7, 0). On separate diagrams, sketch the curve with equation (a) y = f(x) + 3, (3) (b) y = f(2 x). (2) On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the y-axis. ?
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