MATHEMATICS + CURVES

Curves and Transformations

This tutorial examines the sketching of curves. First we examine the sketching of 'primary curves' from which more complicated curves can be drawn using a knowledge of transformations. (Be sure to try the exercises at the end.)

Primary Curves
Listed below are the equations of basic curves that are the simplest to sketch (click on an equation to learn more about it)

• y = mx + b
• y = x2, y = x3, y = x4
• y = 1/x
• y = a^x, y = a^-x
• y = |x|
• y = log_a(x)
• y = a(x � x₁)(x � x₂) ... (x � x_n)
  

Translations

(i) The transformation

       y = f(x) ® y = f(x) + a

translates the graph of y = f(x) by a units up.

In this example the basic curve y = x2 is transformed into y = x2 + a where a ranges from -3 to +3.

(ii) The transformation

       y = f(x) ® y = f(x � a)

translates the graph of y = f(x) across by a units.

In this example the basic curve y = x2 is transformed into y = (x � a)² where a ranges from -3 to +3.

Reflections

(i) The transformation

       y = f(x) ® y = �f(x)

reflects the graph of y = f(x) in the x axis.

In this example the basic curve y = 2^x is transformed into y = �2^x.

(ii) The transformation

       y = f(x) ® y = f(�x)

reflects the graph of y = f(x) in the y axis.

In this example the basic curve y = 2^x is transformed into y = 2^�x.

Stretches and Squeezes

(i) The transformation

       y = f(x) ® y = af(x)

'stretches' the graph of y = f(x) away from the x axis when a > 1. The graph is 'squeezed' towards the x axis when
0 < a < 1.

In this example the basic curve y = x3 is transformed into y = ax3 where a ranges from 0 to +3.

(ii) The transformation

       y = f(x) ® y = f(ax)

'squeezes' the graph of y = f(x) towards the y axis when a > 1. The graph is 'stretched' away from the y axis when
0 < a < 1.

In this example the basic curve
y = (x + 1)(x � 1) is transformed into
y = (ax + 1)(ax � 1) where a ranges from 0 to +3.

Exercises
For each question, sketch the functions together on the one number plane:

1. y = x²,   y = x² + 3

2. y = x³,   y = x³ � 2

3. y = 10^x,   y = �10^x

4. y = log(x),   y = log(x � 2)

5. y = |x|,   y = |x + 1|

6. y = 1/x,   y = �1/x

7. y = |x|,   y = |x � 2|,   y = 3|x � 2|

8. y = x³,   y = (x � 2)³,   y = (x � 2)³ + 1

9. y = 1/x,   y = 2/x,   y = 2/(x-1),   y = 3 + 2/(x-1)

10. y = (x + 1)(x � 1),   y = (x + 1)²(x � 1)³