 Hong Kong
Stage 4 - Stage 5

# Transformations of Functions

Lesson

## Translations of relations

The circle centred on the origin with radius $2$2 has the equation $x^2+y^2=4$x2+y2=4

Suppose we replace the $x$x with $\left(x-15\right)$(x15) and the $y$y with $\left(y-9\right)$(y9) so that our new equation becomes $\left(x-15\right)^2+\left(y-9\right)^2=4$(x15)2+(y9)2=4. The two relations are depicted here. The change from $x$x to $\left(x-15\right)$(x15) and from $y$y to $\left(y-9\right)$(y9) caused the circle to shift to the right by $15$15 units and upward by $9$9 units. The centre moved from $\left(0,0\right)$(0,0) to $\left(15,9\right)$(15,9) and the radius (and thus the overall shape of the circle) remained unchanged.

We can think of the two movements (right and up) as happening independently. The horizontal translation of $15$15 units and the vertical translation of $9$9 units.

## Understanding a translated form

When we think about a function like $y=\left(x+3\right)^2-5$y=(x+3)25, we may consider it as the translation of the basic function $y=x^2$y=x2 (a parabola sitting upright with its minimum turning point at the origin). This means that $y=\left(x+3\right)^2-5$y=(x+3)25, which could be re-written $\left(y+5\right)=\left(x+3\right)^2$(y+5)=(x+3)2, is simply $y=x^2$y=x2 translated $3$3 units to the left, and $5$5 units down.

Something as straight-forward as the linear function $y=x-2$y=x2 can be thought of as the line $y=x$y=x shifted to the right $2$2 units. It could equally be thought of as $y=x$y=x shifted downward by $2$2 units. Can you see why?

Consider more complicated relations like $\frac{\left(x-3\right)^2}{9}+\frac{\left(y+1\right)^2}{4}=1$(x3)29+(y+1)24=1. While we perhaps maybe uncertain about what its basic shape may look like, we understand that its essential form is given by $\frac{x^2}{9}+\frac{y^2}{4}=1$x29+y24=1, with a translation of $3$3 units to the right and $1$1 unit down. In fact the curve is an ellipse.

## An interesting example

A certain curve given by $y=x^2$y=x2 is translated $1$1 unit to the left and $3$3 units upward. In addition to this, the line given by $y=2x-1$y=2x1 is translated $3$3 units to the left.  Where do these translated graphs intersect each other?

We first need to understand that the parabola $y=x^2$y=x2, after translation, will have the form $y-3=\left(x+1\right)^2$y3=(x+1)2. We can write this as $y=\left(x+1\right)^2+3$y=(x+1)2+3. We also see that the line $y=2x-1$y=2x1, after translation, becomes $y=2\left(x+3\right)-1$y=2(x+3)1. After simplification this line has the equation $y=2x+5$y=2x+5

At the intersections of these translated graphs, the $y$y - values are equal, so we can put $\left(x+1\right)^2+3=2x+5$(x+1)2+3=2x+5 and solve for $x$x.

 $\left(x+1\right)^2+3$(x+1)2+3 $=$= $2x+5$2x+5 $x^2+2x+4$x2+2x+4 $=$= $2x+5$2x+5 $x^2-1$x2−1 $=$= $0$0 $\left(x-1\right)\left(x+1\right)$(x−1)(x+1) $=$= $0$0 $x$x $=$= $\pm1$±1

At $x=-1$x=1$y=3$y=3 and at $x=1$x=1$y=7$y=7

The translated graphs and the points of intersection are depicted here: #### Worked Examples

##### Question 1

How do we shift the graph of $y=f\left(x\right)$y=f(x) to get the graph of $y=f\left(x\right)+4$y=f(x)+4?

1. Move the graph up by $4$4 units.

A

Move the graph down by $4$4 units.

B

##### Question 2

How do we shift the graph of $y=g\left(x\right)$y=g(x) to get the graph of $y=g\left(x+6\right)$y=g(x+6)?

1. Move the graph to the left by $6$6 units.

A

Move the graph to the right by $6$6 units.

B

##### Question 3

If the graph of $y=-x^2$y=x2 is translated horizontally $6$6 units to the right and translated vertically $5$5 units upwards, what is its new equation?

##### Question 4

This is a graph of $y=3^x$y=3x.

1. How do we shift the graph of $y=3^x$y=3x to get the graph of $y=3^x-4$y=3x4?

Move the graph $4$4 units to the right.

A

Move the graph downwards by $4$4 units.

B

Move the graph $4$4 units to the left.

C

Move the graph upwards by $4$4 units.

D
2. Hence plot $y=3^x-4$y=3x4 on the same graph as $y=3^x$y=3x.

##### Question 5

This is a graph of $y=\sqrt{4-x^2}$y=4x2.

1. How do we shift the graph of $y=\sqrt{4-x^2}$y=4x2 to get the graph of $y=\sqrt{4-x^2}+2$y=4x2+2?

Move the graph to the right by $2$2 units.

A

Move the graph to the left by $2$2 units.

B

Move the graph downwards by $2$2 units.

C

Move the graph upwards by $2$2 units.

D
2. Hence plot $y=\sqrt{4-x^2}+2$y=4x2+2 on the same graph as $y=\sqrt{4-x^2}$y=4x2.