1. C1: Chapter 4 Graph Sketching
Dr J Frost
Last modified: 21st September 2013

2. 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

3. 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.

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

5. 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

6. 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

7. 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

8. 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.

9. 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

10. 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.

11. 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

12. 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.

13. 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

14. 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

15. 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

16. Exercises
Rest of the questions on your worksheet.