C1: Chapter 4 Graph Sketching Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 21st September 2013

GCSE recap ? ? ? ? ? ? What do the following graphs look like? y = ax2 + bx + c when a > 0 y = ax2 + bx + c when a < 0 y = 1/x y y y ? ? ? x x x y = -3/x y = ax3 + bx2 + cx + d when a > 0 y = ax3 + bx2 + cx + d when a < 0 y ? ? y ? y x x x

GCSE recap – Sketching Quadratics y = x2 – 3x – 4 =(x+1)(x-4) y = 6 – x – x2 = (2-x)(x+3) ? ? y y 6 x x -1 4 -3 2 -4 A ‘sketch’ cares only about the intercepts with the axis and the general shape, not the exact points on the line.

More Examples y = x2 – 4x + 4 =(x-2)2 y = -2x2 + 14x – 24 ? ? y y 4 x x 2 3 4 -24

Sketching Cubics ? ? y = x(x – 1)(x + 1) y = (x – 1)2(x + 2) Is it uphill or downhill? Is the x3 term + or -? Consider the roots: If (x-a) appears once, the line crosses at x = a. If (x-a)2 appears, the line touches at x = a. If (x-a)3 appears, we have a point of inflection at x = a. y = x(x – 1)(x + 1) y = (x – 1)2(x + 2) y y ? ? 2 x x -1 1 -2 1

Sketching Cubics ? ? y = x2(2 – x) y = (x – 1)3 Is it uphill or downhill? Is the x3 term + or -? Consider the roots: If (x-a) appears once, the line crosses at x = a. If (x-a)2 appears, the line touches at x = a. If (x-a)3 appears, we have a point of inflection at x = a. y = x2(2 – x) y = (x – 1)3 ? y ? y x x -1 2 1 -1

Exercises Sketch the following, ensuring you indicate the values where the line intercepts the axes. y = (3-x)3 1 y = (x+2)(x-1)(x-3) 5 y = x(x+1)2 9 ? ? ? 27 6 3 -2 1 3 -1 2 y = x(x-1)(2-x) 6 y = x(1 – x)2 10 y = (x+2)2(x-1) ? ? ? 1 2 -2 1 1 -4 7 y = -x3 11 y = (2-x)(x+3)2 3 y = x(2x – 1)(x + 3) ? ? ? 18 0.5 3 -3 2 4 y = x2(x + 1) 8 y = (x+2)3 12 y = (1 – x)2(3 – x) ? ? ? 8 3 -1 -2 1 3

Transforming Graphs – GCSE Recap Suppose we sketch the function y = f(x). What happens when we sketch each of the following? f(x + 3)  3 ? f(x – 2)  2 ? f(2x)  Stretch x by factor of ½ ? ↔ Stretch x by factor of 3 f(x/3) ? ↑4 f(x) + 4 ? 3f(x) ↕ 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.

a f(bx + c) + d Transforming Graphs – GCSE Recap Step 1: Step 3: Bro Tip: To get the order of transformations correct inside the f(..), think what you’d need to do to get from (bx + c) back to x. Step 1: ?  c Step 3: ? ↕ a Step 4: ? ↑ d Step 2: ? ↔  b

Quickfire Questions 2f(2x – 1) f(0.5x + 1) - 2 f(-x) -2f(-2x + 3) + 1 List the transformations required (in order). 2f(2x – 1) f(0.5x + 1) - 2 ? ? Shift right 1 unit. Halve x values. Double y values. Shift left 1 unit. Double x values. Shift down 2 units. f(-x) -2f(-2x + 3) + 1 ? ? Times x values by -1, i.e. reflect in y-axis. Shift left 3 units. Divide x values by -2 (i.e. Halve and reflect in y-axis. Times y values by -2, i.e. Reflect in x axis and double y. Shift up 1 unit.

f(-x) vs –f(x) We don’t have to reason about these any differently! y = f(x) y (2, 3) 1 x y = -1 y = f(-x) y = -f(x) ? y ? y Change inside f brackets, so times x 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

Exercise ? ? ? y = f(x) y = 2f(x+2) y = -f(-x) – 1 y = f(2x) Here is the graph y = f(x). Draw the following graphs, ensuring you indicate where the graph crosses the coordinate axis, minimum/maximum points, and the equations of any asymptotes. y y = f(x) (2, 3) 1 x y = 2f(x+2) y = -1 ? 6 x y y = -2 y = -f(-x) – 1 ? -2 x y y = 0 y = f(2x) ? 1 x y y = -1 (1, 3) (-2, -4) Then try Q5 + 7 on the provided worksheet.

Drawing transformed graphs Bro Tip: To sketch many functions, it’s best to start with a similar simpler function (in this case 𝑓 𝑥 = 𝑥 3 ), then consider how it’s been transformed. Sketch y = (x – 1)3 + 8 y ? 7 -1 x

Drawing transformed graphs Sketch 𝒚= 𝟏 𝒙+𝟐 −𝟏 (Hint: If f(x) = 1/x, then what is the above function?) ? y -2 𝑦=−1 𝑥=−1 -0.5 x

Exercises Sketch the following, ensuring you indicate the points at which the lines cross the coordinate axis, and the equations of any asymptotes. Q1 𝑦= 1 𝑥+3 +4 Q2 𝑦=− 2 𝑥−1 ? y ? y 13 3 y = 4 x = 1 2 x 13 4 - x x = -3

Exercises Rest of the questions on your worksheet.