Informative line

Reflection Transformations

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

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.

Illustration Questions

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

A

B

C

D

×

Consider the graph g(x) = cos x

image

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

A image
B image
C image
D image

Option A is Correct

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 \(x\).

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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 image
B image
C image
D image

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).

Illustration Questions

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 \(f(-x)\)?

image
A image
B image
C image
D image

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 the entire graph in x-axis. (x-axis acts as a mirror)

Illustration Questions

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.

image
A image
B image
C image
D image

Option B is Correct

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