Transformations

Horizontally or Vertically or both Translations applied to a graph mean move movements Horizontally or Vertically or both They can be described using a vector as shown below Means translate(move) a graph 5 to the right And translate (move it) 2 up

The graph of forms a curve called a parabola This point . . . is called the vertex

Adding a constant translates up the y-axis e.g. The vertex is now ( 0, 3) has added 3 to the y-values

Adding 3 to x gives We get Adding 3 to x moves the curve 3 to the left.

We can write this in vector form as: translation Translating in both directions e.g. We can write this in vector form as: translation

SUMMARY The curve is a translation of by The vertex is given by

SUMMARY Memory Aid HIVO – Horizontal Inside the Bracket Vertical Outside the bracket HOVIS – Horizontal Opposite of what it says Vertical Is Same as what it says

Exercises: Sketch the following translations of 1. 2. 3.

4 Sketch the curve found by translating by . What is its equation? 5 Sketch the curve found by translating by . What is its equation?

e.g. The translation of the function by the vector gives the function . The graph becomes Means translate horizontally by +2. So put –2 in the bracket. HIvo HOvis

e.g. The translation of the function by the vector gives the function . The graph becomes Means translate vertically by +1. So put +1 outside the bracket. HIVO HOVIS

Vertical Stretches e.g.1 Consider the following functions: and For For In transforming from to the y-value has been multiplied by 4

e.g.1 Consider the following functions: and For For In transforming from to the y-value has been multiplied by 4 Similarly, for every value of x, the y-value on is 4 times the y-value on is a stretch of scale factor 4 parallel to the y-axis

e.g.1 Consider the following functions: and HIVO – Horizontal Inside the Bracket Vertical Outside the bracket HOVIS – Horizontal Opposite of what it says Vertical Is Same as what it says As the 4 is Outside the stretch is applied Vertically

The graphs of the functions are as follows: is a stretch of by scale factor 4, parallel to the y-axis

is a transformation of given by a stretch of scale factor 4 parallel to the y-axis

3 1 Horizontal stretches Now, for and for The x-value must be divided by 3 to give the same value of y. So the x coordinate is stretched by a factor of  parallel to the x axis

e.g.1 Consider the following functions: and HIVO – Horizontal Inside the Bracket Vertical Outside the bracket HOVIS – Horizontal Opposite of what it says Vertical Is Same as what it says As the 3 is Inside the stretch is applied Horizontally and is the Opposite. So instead of being 3 times as wide it is a  as wide.

SUMMARY The transformation of to is a stretch of scale factor 4 parallel to the y-axis or is a stretch of scale factor parallel to the x-axis

SUMMARY The function is obtained from by a stretch of scale factor ( s.f. ) k, parallel to the y-axis. The function is obtained from by a stretch of scale factor ( s.f. ) , parallel to the x-axis.

e.g. 2 Describe the transformation of that gives . Using the same axes, sketch both functions. Solution: can be written as so it is a stretch of s.f. 3, parallel to the y-axis We always stretch from an axis.

Exercises 1. (a) Describe a transformation of that gives . (b) Sketch the graphs of both functions to illustrate your answer. Solution: (a) A stretch of s.f. 9 parallel to the y-axis. (b)

Transforming the exponential graph y = ex -1 -2 -3 -4 -5 1 2 3 4 5

Horizontal stretch scale factor  All the x coordinates are  as wide y = ex y = e(2x) -1 -2 -3 -4 -5 1 2 3 4 5 -1 -2 -3 -4 -5 1 2 3 4 5 Horizontal stretch scale factor  All the x coordinates are  as wide

Horizontal stretch scale factor  Vertical stretch scale factor 2 y = ex y = 2e(2x) -1 -2 -3 -4 -5 1 2 3 4 5 Horizontal stretch scale factor  Vertical stretch scale factor 2 All the y coordinates are 2x as high

y = ex y = 2e(2x)–4 Horizontal stretch scale factor  Vertical stretch scale factor 2 Vertical translation

Two more Transformations Reflection in the x-axis Every y-value changes sign when we reflect in the x-axis e.g. y=x2 y=–(x)2 So, In general, a reflection in the x-axis is given by

Reflection in the y-axis Every x-value changes sign when we reflect in the y-axis e.g. In general, a reflection in the y-axis is given by

SUMMARY Reflections in the axes Reflecting in the x-axis changes the sign of y Reflecting in the y-axis changes the sign of x