TRANSFORMATIONS

TIP: When doing these sort of problems, remember:

a) Function notation is used as a shorthand to show how curves can be transformed in a general way. Instead of saying the equation of a line is x2 -4x + 3 etc. we just say that y = f(x) or y is a function of x. Just think of a function as being a "black box" where you pass it input and it gives you output. e.g. If f(x) = x + 3, then f(x +2) = (x + 2) + 3, -f(x) = -x - 3 etc.

b) The graph y = f(x) is mapped to y = f(x + a) by moving the graph left by a along the x-axis. f(x - a) moves the graph right by a along the x-axis.

c) The graph y = f(x) is mapped to y = f(x) + b by moving the graph up by b along the y-axis.

d) The graph y = f(x) is mapped to y = -f(x) by reflection in the x-axis.

e) The graph y = f(x) is mapped to y = f(-x) by reflection in the y-axis.

e) The graph y = f(x) is mapped to y = f(hx) by a one-way stretch i.e. multiply x coordinate by 1/h.

f) The graph y = f(x) is mapped to y = kf(x) by a one-way stretch i.e. multiply y coordinate by k.

Solve the following:

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)

 

2. Describe the transformations which transform the graph
y = (x + 1)2 to the following:

a) y = (x + 5)2

b) y = (x - 2)2 - 6

 

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.

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)

 

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)

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.

7. Sketch the graphs of:

a) y = sin x

b) y = -sin x

c) y = -sin 2x

d) y = 1 - sin 2x

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.

Ok Here's the Answers