1. The function f(x) = x² + 3x - 1. Find the values of the functions below:

a) f(x + 2)

b) f(-3x)

c) -f(2x)

a) All you do is subtitute x + 2 in the f(x) equation, giving:

(x + 2)² + 3(x + 2) - 1

= (x + 2)(x + 2) + 3(x + 2) - 1

= x² + 4x + 4 + 3x + 6 - 1

= x² + 7x + 9

b) Similarly,

(-3x)² + 3(-3x) - 1

= 9x² - 9x - 1

c) and again, here we use square brackets also to remind us that we need to apply the negative sign to the whole of the answer.

- [ (2x)² + 3(2x) - 1]

= - [ 4x² + 6x - 1]

= -4x² - 6x + 1

a) If f(x) = (x + 1)² then f(x + 4) = (x + 5)², so we are looking at moving 4 to the left on the x-axis (in maths speak, this represents a horizontal translation of 4 units in the negative x direction).

b) This is f(x -3) - 6, which means move 3 to the right on the x-axis, then move down the y-axis 6. (or maths speak again: a horizontal translation of 3 units in the positive x direction, followed by a vertical translation of 6 units in the negative y direction).

3. If the transformations indicated below are applied to the graph
y = x² - 9x, what is the equation of the new graph?

A horizontal translation of 2 units in the positive x direction followed by a vertical translation of 1 unit down.

A horizontal translation means "move along the x-axis", in the positive direction means "to the right". This transformation should be followed by moving 1 unit down the y-axis.

i.e. the first transformation is y = f(x - 2) and the second transformation is y = f(x) - 1. This gives us:

y = (x - 2)² - 9(x -2) - 1

y = (x - 2)(x - 2) - 9(x - 2) - 1

y = x² - 4x + 4 - 9x + 18 - 1

y = x² - 13x + 21

4. Here is a sketch of a curve with the equation y = f(x):

The vertex (top) of the curve is at position (3, 10). Write down the coordinates of the vertex for each of the curves below:

a) y = f(x) + 2

b) y = f(x + 4)

c) y = f(3x)

d) y = f(-x)

a) This means move 2 up on the y-axis, so you need to add 2 to the y coordinate in (3, 10) giving (3, 12).

b) This means move 4 to the left on the x-axis giving
(-1, 10).

c) This means multiply x by 1/3 and leave y unchanged giving: (1, 10).

d) This means reflect in the y-axis (picture what will happen here) and you'll find that you get (-3, 10).

5. Where f(x) = 2 / x, sketch the graph of y = f(x).

Sketch the following:

a) y = f(2x)
b) y = f(x) + 1
c) y = f(x + 1)
d) y = ½ f(x)
e) y = f(1 / x)

The graph is a classic shape for a "reciprocal" function (i.e. a function where x is at the bottom of the division line):

a) f(2x) =  2    =   1   (x is reduced by ½)
                2x        x

b) f(x) + 2 = 2  + 1 (graph moves up 1 unit)
                    x

c) f(x + 2) =    2    (graph moves to left by 2 units)
                    x + 2

d) ½ f(x) =   1  (exactly the same graph as for question a)
                    x

e) f(1/x) =     2   = 2x
                  (1/x)

                   

6. Here is a sketch of the graph y = f(x) where -3 £ x ³ 3:

a) Draw the graph of y = f(x) + 2.

b) Draw the graph of y = f(x - 3).

c) Draw the graph of y = f(-x) + 1.

a) f(x) + 2 means move the graph 2 units up the y-axis.

b) f(x - 3) means move the move 3 units to the right on the x-axis.

c) f(-x) + 1 means reflect in the y-axis and move the graph 1 unit up the y-axis.

7. Where -360^o £ x ³ 360^o Sketch the graphs of:

a) y = sin x

b) y = -sin x

c) y = -sin 2x

d) y = 1 - sin 2x

a) If you remember to split 360^o into quarters, even if you can't remember how this classic graph is drawn, quickly work out the values for sin -360, -270, -180, -90, 0, 90, 180, 270, 260 on your calculator and you'll find that they are all 1, -1 or 0 which makes the graph very easy to sketch:

a) This is a standard sine curve:

b) y = -f(x) i.e. reflect in the x-axis:

c) y = -f(2x) i.e. reduce x by ½ and then reflect in x-axis:

d) y = -f(2x) + 1 i.e. move the graph already calculated in question c) 1 unit up the y-axis:

8. The diagram shows the curve with the equation y = f(x)
where f(x) = x² - 5x + 4.

a) Sketch the curve with the equation y = f(x - 2).

b) The curve with equation y = f(x) intersects with the curve with equation y = f(x - a) at the point P. Give the x-coordinate at the point P in terms of a.

c) The curve y = x² - 5x + 4 is reflected in the y-axis, find the equation of this new curve.

a) y = f(x - 2) means move 2 units right on the x-axis:

b) At point P f(x) = f(x - a)

f(x) = x² - 5x + 4

f(x + a) = (x + a)² - 5(x + a) + 4

So (remember your algebra here),

x² - 5x + 4 = (x + a)² - 5(x + a) + 4

x² - 5x + 4 = x² - 2ax + a² -5x + 5a + 4

Cancel out bits which are the same on both sides:

0 = -2ax + a² + 5a

2ax = a² + 5a

x = a² + 5a
          2a 

c) A reflection in the y-axis means y = f(-x)

y = (-x)² - 5(-x) + 4

y = x² + 5x + 4