IGCSE FMC 1 Sketching Graphs Dr J Frost

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IGCSE FM/C 1 Sketching Graphs Dr J Frost (jfrost@tiffin. kingston. sch. uk) Objectives: (from

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

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

#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

#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 ?

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

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

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?

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

Understanding features of a quadratic IGCSE FM June 2012 Paper 2 Q 4 ? ? ?

 RECAP : : Using completed square for min/max ? ? ?

RECAP : : Using completed square for min/max ? ? ?

 Write down !

Write down !

 Complete the table, and hence sketch the graphs Equation Completed Square x at

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 1 (Exercises on provided sheet)

 Exercise 1 2 (Exercises on provided sheet) 3 a ? ? b ?

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) 5 4 ? ?

 Exercise 1 (Exercises on provided sheet) 6 7 ? ?

Exercise 1 (Exercises on provided sheet) 6 7 ? ?

#2 b : : Sketching Cubics A recap of their general shape from GCSE…

#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 b : : Sketching Cubics y y 2 ? -1 1 x ? -2 1 x

More Examples y y ? -1 2 x 1 -1 ? x A point

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

Test Your Understanding ? -3 4 ? -1 2 ? -3 3

Quickfire Questions! Sketch the following, ensuring you indicate the values where the line intercepts

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

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) 3 4 ? ? ?

 Exercise 2 (Exercises on provided sheet) 5 ?

Exercise 2 (Exercises on provided sheet) 5 ?

 Exercise 2 6 (Exercises on provided sheet) Suggest equations for the following cubic

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 ?

Exercise 2 (Exercises on provided sheet) 7 ?

 #2 c : : Reciprocal Graphs We’ll be able to sketch more complicated

#2 c : : Reciprocal Graphs We’ll be able to sketch more complicated graphs of this form:

 Example ?

Example ?

 Example ?

Example ?

 Test Your Understanding ?

Test Your Understanding ?

 Exercise 3 1 ? ?

Exercise 3 1 ? ?

 Exercise 3 2 ?

Exercise 3 2 ?

 Exercise 3 3 ?

Exercise 3 3 ?

 Exercise 3 4 ?

Exercise 3 4 ?

 #3 : : Piecewise Functions Sketch > (2, 9) (0, 5) (-1, 0)

#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,

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 ? ?

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) 4 3 Sketch ? ? ?

 Exercise 4 (Exercises on provided sheet) 6 5 2 1 2 3 4

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

#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

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

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 7 -1 ? x

Drawing transformed graphs y -2 -0. 5 ? x

Drawing transformed graphs y -2 -0. 5 ? x

Test Your Understanding C 1 June 2009 Q 10 b) a ? b ?

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) 1 ? ?

Exercise 5 (Exercises on provided sheet) 2 [C 1 May 2010 Q 6] Figure

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]

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]

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) 5 ?

Exercise 5 (Exercises on provided sheet) 6 ? [C 1 June 2008 Q 3]

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. ?