NZ Level 7 (NZC) Level 2 (NCEA) Transformations of Logarithmic graphs (y=klogx+c)
Lesson

## Vertical translation

We have previously looked at certain transformations of functions. These transformations include scaling, translating, and reflecting, all of them in either the horizontal or vertical direction. These transformations can all be applied to logarithmic functions in the same way.

In particular, recall that adding a constant to the function corresponds to translating the graph vertically. So the graph of $g\left(x\right)=\log_bx+k$g(x)=logbx+k is a vertical translation of the graph of $f\left(x\right)=\log_bx$f(x)=logbx. The translation is upwards if $k$k is positive, and downwards if $k$k is negative. Graphs of $f\left(x\right)=\log_bx$f(x)=logbx and $g\left(x\right)=\log_bx+k$g(x)=logbx+k, for $k<0$k<0.

Notice that the asymptote is not changed by a vertical translation, and is still the line $x=0$x=0. The $x$x-intercept has changed however, and now occurs at a point further along the $x$x-axis. The original $x$x-intercept (which was at $\left(1,0\right)$(1,0)) has now been translated vertically to $\left(1,k\right)$(1,k) and is no longer on the $x$x-axis.

Let's look at an example involving a horizontal reflection too.

#### Worked example

##### example 1

The graphs of the function $f\left(x\right)=\log_3\left(-x\right)$f(x)=log3(x) and another function $g\left(x\right)$g(x) are shown below. • Describe the transformation used to get from $f\left(x\right)$f(x) to $g\left(x\right)$g(x).

Think: $g\left(x\right)$g(x) has the same general shape as $f\left(x\right)$f(x), just translated upwards. We can figure out how far it has been translated by looking at the distance between corresponding points.

Do: The point on $g\left(x\right)$g(x) that is directly above the $x$x-intercepts of $f\left(x\right)$f(x) is at $\left(-1,5\right)$(1,5), which is $5$5 units higher. In fact, we can see the constant distance of $5$5 units all the way along the function: So $f\left(x\right)$f(x) has been translated $5$5 units upwards to give $g\left(x\right)$g(x).

• Determine the equation of the function $g\left(x\right)$g(x).

Think: We know that $f\left(x\right)$f(x) has been vertically translated $5$5 units upwards to give $g\left(x\right)$g(x). That is, the function has been increased by $5$5.

Do: This means that $g\left(x\right)=\log_3\left(-x\right)+5$g(x)=log3(x)+5. This function has an asymptote at $x=0$x=0, and the negative coefficient of $x$x means that it takes values to the left of the asymptote, just like $f\left(x\right)$f(x).

Functions of the form $y=\log_b\left(\pm x\right)+k$y=logb(±x)+k

A function of the form $y=\log_bx+k$y=logbx+k represents a vertical translation by $k$k units of the function $y=\log_bx$y=logbx.

Similarly, a function of the form $y=\log_b\left(-x\right)+k$y=logb(x)+k represents a vertical translation by $k$k units of the function $y=\log_b\left(-x\right)$y=logb(x).

In both cases:

• The translation is upwards if $k$k is positive, and downwards if $k$k is negative.
• The asymptote of the translated function remains at $x=0$x=0.

#### Practice questions

##### Question 1

A graph of the function $y=\log_2x$y=log2x is shown below.

A graph of the function $y=\log_2x+2$y=log2x+2 can be obtained from the original graph by transforming it in some way.

1. Complete the table of values below for $y=\log_2x$y=log2x:

 $x$x $\frac{1}{2}$12​ $1$1 $2$2 $4$4 $\log_2x$log2​x $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

2. Now complete the table of values below for $y=\log_2x+2$y=log2x+2:

 $x$x $\frac{1}{2}$12​ $1$1 $2$2 $4$4 $\log_2x+2$log2​x+2 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

3. Which of the following is a graph of $y=\log_2x+2$y=log2x+2?

A

B

C

D

A

B

C

D
4. Which features of the graph are unchanged after it has been translated $2$2 units upwards?

Select all that apply.

The vertical asymptote.

A

The general shape of the graph.

B

The $x$x-intercept.

C

The range.

D

The vertical asymptote.

A

The general shape of the graph.

B

The $x$x-intercept.

C

The range.

D

##### Question 2

Which of the following options shows the graph of $y=\log_3x$y=log3x after it has been translated $2$2 units up?

A

B

C

D

A

B

C

D

##### Question 3

The function $y=\log_5x$y=log5x is translated downwards by $2$2 units.

1. State the equation of the function after it has been translated.

2. The graph of $y=\log_5x$y=log5x is shown below. Draw the translated graph on the same plane.

## Dilation

We have previously looked at certain transformations of functions. These transformations include scaling, translating, and reflecting, all of them in either the horizontal or vertical direction. These transformations can all be applied to logarithmic functions in the same way.

In particular, recall that multiplying a function by a constant corresponds to vertically rescaling the function. The graph of $g\left(x\right)=a\log_bx$g(x)=alogbx is a vertical dilation of the graph of $f\left(x\right)=\log_bx$f(x)=logbx if $\left|a\right|$|a| is greater than $1$1, and a vertical compression if $\left|a\right|$|a| is between $0$0 and $1$1. ### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems