Informative line

### Reflection Transformations

Learn how to do reflections functions in transformations of graphs. Practice reflection of graph in y-axis.

# Negative Modulus Function

• We define modulus function

$$f(x)=|x|=\begin {cases}{x\,\,\,\, if\,\,\, x \geq0\\-x\,\,\,\,\,if\,\,\,\,\,x<0}\end {cases}$$

The idea of the function is to return the absolute value of the input quantity $$x$$

$$\therefore \;|5|=5$$ and $$|-6|=-(-6)=6$$

• The above is the graph of $$f(x)=|x|$$ v/s $$x.$$It resembles $$y=x$$ line for $$x\geq 0$$ and $$y=-x$$ line for $$x<0$$.
• The negative modulus function is defined as

$$f(x)=-|x|=\begin {cases}{-x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, if\,\,\, x \geq0\\-(-x)=x\,\,\,\,\,if\,\,\,\,\,x<0}\end {cases}$$

The above is the graph of $$f(x)=-|x|$$ v/s $$x$$.

#### If $$f(x)=-|2x+7|$$ then the value of $$f(-3)$$ is

A –4

B 1

C –1

D 5

×

$$-|x|=\begin {cases}{-x\,\,\,\, if\,\,\, x \geq0\\\,\,\,x\,\,\,\,\,if\,\,\,\,\,x<0}\end {cases}$$

In this case

$$f(x)=-|2x+7|$$

$$\therefore f(-3)=-|2×(-3)+7|=-|1|=-1$$

$$\therefore f(-3)=-1$$

### If $$f(x)=-|2x+7|$$ then the value of $$f(-3)$$ is

A

–4

.

B

1

C

–1

D

5

Option C is Correct

# Difference between the Transformation  f(-x) and  -f(x) on a Certain Function

• To obtain $$-f(x)$$ from $$f(x)$$ we find the value of the function at particular $$x$$ and then negate the value obtained while in $$f(-x)$$, we negate the value of $$x$$ first and then find the value of the  function at that negated value.
• Sketch of  $$-f(x)$$  is obtained by taking reflection of graph of $$f(x)$$ about  $$x$$ axis.
• Sketch of  $$f(-x)$$  is obtained by taking reflection of graph of $$f(x)$$ about  $$y$$ axis.

#### If $$f(x)=3\,sin\,2x$$, find the value of $$-4f(x)+3f(-x)$$ at $$x=\dfrac {\pi}{4}$$.

A 11

B –21

C 20

D –15

×

$$-f(x)=-$$(value of $$f(x)$$)

$$f(-x)=f$$ (negative $$x$$)

In this case,

$$f(x)=3\,sin\,2x$$

$$\therefore -4f(x)+3f(-x)=-4f\left ( \dfrac {\pi}{4}\right)+3f\left ( \dfrac {-\pi}{4}\right)$$ at $$x=\dfrac {\pi}{4}$$

$$=-4×3\,sin\, \dfrac {\pi}{2}+3×3\,sin\, \Big( \dfrac {-\pi}{2}\Big)$$

$$=-12-9=-21$$

### If $$f(x)=3\,sin\,2x$$, find the value of $$-4f(x)+3f(-x)$$ at $$x=\dfrac {\pi}{4}$$.

A

11

.

B

–21

C

20

D

–15

Option B is Correct

# Graphical Transformation (Identification of transformation)

## Other transformations (Stretching and reflecting)

Suppose we know the graph of $$f(x)$$ then graph of  following function can be constructed using some appropriate transformation on $$'f'$$.

S.No. Function Transformation on $$f(x)$$
1 $$y=c\,f(x)\;\;(c>0)$$ Stretch vertically by a factor of 'c'.
2 $$y=c\,f(x)\;\;(0<c<1)$$ Shrink vertically by a factor of 'c'.
3 $$y=f(c\,x)\;\;(c>1)$$ Shrink horizontally by a factor of 'c'.
4 $$y=f(c\,x)\;\;(0<c<1)$$ Stretch horizontally by a factor of 'c'.
5 $$y=-f(x)$$ Take reflection about $$x$$-axis.
6 $$y=f(-x)$$ Take reflection about $$y$$-axis.

e.g.

## Graphical Transformation

If we know the graph of a certain function say $$f(x)$$ then we can sketch the graphs of many functions which are related to $$'f'$$ by making some appropriate changes in the graph of $$'f'$$.

The following tables show the function on the left and the transformation or change required to be done to the graph of $$'f'$$ obtain the graph of this function.

S.No. Function Transformation on $$f(x)$$
1 $$y=f(x)+c\;\;(c>0)$$ Shift the graph 'c' units upwards
2 $$y=f(x)-c\;\;(c>0)$$ Shift the graph 'c' units downwards
3 $$y=f(x-c)\;\;(c>0)$$ Shift the graph 'c' units to right
4 $$y=f(x+c)\;\;(c>0)$$ Shift the graph 'c' units to left.

e.g.

We have shifted the graph of$$f(x)$$ by 1 units in the upward direction.

#### The given option show a pair of graph one of which is obtained by an appropriate transformation on the other, the name of transformation is given, identify the correct option

A

B

C

D

×

S.No. Function Transformation on $$f(x)$$
1 $$y=c\,f(x)\;\;(c>0)$$ Stretch vertically by a factor of 'c'.
2 $$y=c\,f(x)\;\;(0<c<1)$$ Shrink vertically by a factor of 'c'.
3 $$y=f(c\,x)\;\;(c>1)$$ Shrink horizontally by a factor of 'c'.
4 $$y=f(c\,x)\;\;(0<c<1)$$ Stretch horizontally by a factor of 'c'.
5 $$y=-f(x)$$ Take reflection about $$x$$-axis.
6 $$y=f(-x)$$ Take reflection about $$y$$-axis.

S.No. Function Transformation on $$f(x)$$
1 $$y=f(x)+c\;\;(c>0)$$ Shift the graph 'c' units upwards
2 $$y=f(x)-c\;\;(c>0)$$ Shift the graph 'c' units downwards
3 $$y=f(x-c)\;\;(c>0)$$ Shift the graph 'c' units to right
4 $$y=f(x+c)\;\;(c>0)$$ Shift the graph 'c' units to left.

### The given option show a pair of graph one of which is obtained by an appropriate transformation on the other, the name of transformation is given, identify the correct option

A
B
C
D

Option C is Correct

# Graph of f(-x) when graph of f(x) is known or given

• To sketch the graph of $$f(-x)$$ when graph of $$f(x)$$ is known or given, we take the reflection of entire graph in y-axis (y-axis acts as a mirror).

#### The graph of a function $$'f'$$ is as shown . Which of the following represents the graph of $$f(-x)$$?

A

B

C

D

×

For the graph of $$f(-x)$$ take reflection about y-axis, we can also observe a particular point $$(h,k)$$ on the graph of $$'f'$$ and look for $$(-h,k)$$ in the required graph.

$$\therefore$$ Hence option 'C' is correct.

### The graph of a function $$'f'$$ is as shown . Which of the following represents the graph of $$f(-x)$$?

A
B
C
D

Option C is Correct

# Modulus Transformation

• To sketch three graph of |f(x)| from the graph of 'f', we retain the part of the graph which is above x-axis as it is (|f(x)| = f(x) if $$f(x) \geq 0$$) and part below x-axis is reflected about x-axis.

e.g.

#### Which of the following is the graph of f(x) = |cos x| ?

A

B

C

D

×

Consider the graph g(x) = cos x

### Which of the following is the graph of f(x) = |cos x| ?

A
B
C
D

Option A is Correct

# Graph of -f(x) when graph of f(x) is known or given

• To sketch the graph of $$-f(x)$$ when graph of $$f(x)$$ is known or given, we take the reflection of the entire graph in x-axis. (x-axis acts as a mirror)

#### Let $$'f'$$ be a function whose graph is shown.

A

B

C

D

×

The reflection of $$'f'$$ in x-axis. You can also select a particular point on the graph of $$'f'$$ and take its reflection in x-axis. ( The reflection of $$(h,k)$$ will be $$(h,-k)$$. The reflected point should be a part of graph of $$-f(x)$$ .

$$\therefore$$ option 'B'  $$(-2, -3)$$ is reflected to $$(-2, 3)$$.

### Let $$'f'$$ be a function whose graph is shown.

A
B
C
D

Option B is Correct