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
?
?
x
y = -3/x
y
?
y = ax2 + bx + c when a < 0
y
x
y = 1/x
y
?
x
x
y = ax3 + bx2 + cx + d when a > 0
y = ax3 + bx2 + cx + d when a < 0
y
y
?
?
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
?
-1
?
4
x
-3
-4
A ‘sketch’ cares only about the intercepts
with the axis and the general shape, not the
exact points on the line.
2
x
More Examples
y = x2 – 4x + 4
=(x-2)2
y = -2x2 + 14x – 24
= 2(3-x)(x-4)
y
y
4?
?
2
x
3
-24
4
x
Sketching Cubics
1. Is it uphill or downhill? Is the x3 term + or -?
2. Consider the roots:
a) If (x-a) appears once, the line crosses at x = a.
b) If (x-a)2 appears, the line touches at x = a.
c) If (x-a)3 appears, we have a point of inflection at x = a.
y = (x – 1)2(x + 2)
y = x(x – 1)(x + 1)
y
y
2
?
-1
1
x
?
-2
1
x
Sketching Cubics
1. Is it uphill or downhill? Is the x3 term + or -?
2. Consider the roots:
a) If (x-a) appears once, the line crosses at x = a.
b) If (x-a)2 appears, the line touches at x = a.
c) If (x-a)3 appears, we have a point of inflection at x = a.
y = x2(2 – x)
y = (x – 1)3
y
y
?
-1
2
?
x
1
-1
x
Exercises
Sketch the following, ensuring you indicate the values where the line intercepts the axes.
1
y = (x+2)(x-1)(x-3)
5
y = x(x+1)2
9
y = (3-x)3
27
6
-2
2
?
1
y = x(x-1)(2-x)
?
1
6
?
?
-1
3
y = x(1 – x)2
?
2
3
10
y = (x+2)2(x-1)
?
1
-2
1
-4
3
y = x(2x – 1)(x + 3)
7
y = -x3
11
y = (2-x)(x+3)2
18
?
0.5
4
y = x2(x + 1)
?
3
8
y = (x+2)3
12
8
-1
?
-2
?
-3
2
y = (1 – x)2(3 – x)
3
?
?
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
f(x/3)
↔ Stretch x by factor
of 3
?
f(x) + 4
↑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.
Transforming Graphs – GCSE Recap
a f(bx + c) + d
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 2:
↔
?b
Step 4:
↑d ?
Quickfire Questions
List the transformations required (in order).
2f(2x – 1)
•Shift right 1 unit.
•Halve x ?
values.
•Double y values.
f(-x)
•Times x values by -1,
i.e. reflect
? in y-axis.
f(0.5x + 1) - 2
•Shift left 1 unit.
•Double x values.
?
•Shift down 2 units.
-2f(-2x + 3) + 1
•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)
1
y=1
x
?
-1
?
x
y = -1
(2, -3)
Change outside f
brackets, so times
y values by -1
Exercise
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
y
6
y = -f(-x) – 1
?
x
y
y = f(2x)
y
y = -2
y=0
x
?
(1, 3)
1
-2
?
(-2, -4)
x
y = -1
Then try Q5 + 7 on the
provided worksheet.
Drawing transformed graphs
Sketch y = (x –
1)3
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.
+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
𝑥 = −1
-2
?
x
-0.5
𝑦 = −1
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
+4
𝑥+3
Q2
𝑦=−
y
2
𝑥−1
y
13
3
y=4
2
x
?
x=1
x = -3
- 13
4
?
x
Exercises
Rest of the questions on your worksheet.