Transformations of Functions - Practice Questions & MCQ

$f(x) \rightarrow a f(x), a>1$

Stretching of a graph along the y-axis occurs if we multiply all outputs y of a function by the same positive constant (here 'a' ). 

$f(x) \rightarrow \frac{1}{a} f(x)$      (a > 1)

Shrinking of a graph along the y-axis occurs if we multiply all outputs y of a function by the same positive constant (here '1/a' ). 

For Example :

the graph of the function $f(x)=3 x^2$ is the graph of $y=x^2$ stretched vertically by a factor of 3, whereas the graph of $\mathrm{f}(\mathrm{x})=\frac{1}{3} \mathrm{x}^2$  is the graph of y=x² compressed vertically by a factor of 3. 

$f(x)$ transforms to $f(a x) \quad(a>1)$
Shrink the graph of $f(x)$ 'a' times along the $x$-axis after drawing the graph of $f(x)$,
$f(x)$ transforms to $f(x / a) \quad(a>1)$
Stretch the graph of $\mathrm{f}(\mathrm{x})$ ' a ' times along the x -axis after drawing the graph of $\mathrm{f}(\mathrm{x})$,
For Example: The graph of $f(x)=\sin x, f(x)=\sin (2 x)$, and $f(x)=\sin (x / 2)$.

Transformation $\mathrm{f}(\mathrm{x}) \rightarrow \mathrm{f}(-\mathrm{x})$,
When we multiply all inputs by -1 , we get a reflection about the $y$-axis
So, to draw $\mathrm{y}=\mathrm{f}(-\mathrm{x})$, take the image of the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ in the y -axis as a plane mirror $f(x) \rightarrow-f(x):$

When we multiply all the outputs by -1 , we get a reflection about the x -axis.
To draw $y=-f(x)$ take an image of $f(x)$ in the $x$-axis as a plane mirror
For example
The graph of $y=e^x, y=-e^x \quad$ (Transformation $\left.f(x) \rightarrow-f(x)\right)$

$
f(x) \rightarrow|f(x)|
$
When $y=f(x)$ given
Leave the positive part of $f(x)$ (the part above the $x$-axis) as it is

Now, take the image of the negative part of $f(x)$ (the part below the $x$-axis) about the $x$-axis.
OR

Take the mirror image in the $x$-axis of the portion of the graph of $f(x)$ which lies below the $x$-axis

For Example: 

                y=x³                                                               y=|x³|                                             y=|x³| and y=x³

Transformation $\mathrm{f}(\mathrm{x}) \rightarrow \mathrm{f}(|\mathrm{x}|) \mid$
When $y=f(x)$ given
Leave the graph lying right side of the $y$-axis as it is
The part of $f(x)$ lying on the left side of the $y$-axis is deleted.
Now, on the left of the $y$-axis take the mirror image of the portion of $f(x)$ that lying on the right side of the $y$-axis.

For Example:

                 y = f(x)                                     y = f(x) and y = f(|x|)                                 y = f(|x|)                                      

Transformation $\mathrm{f}(\mathrm{x}) \rightarrow|\mathrm{f}(|\mathrm{x}|)|$
First $\mathrm{f}(\mathrm{x})$ is transforms to $|\mathrm{f}(\mathrm{x})|$
Then $|f(x)|$ transforms to $|\mathrm{f}(|\mathrm{x}|)|$

Or

(i) $f(x) \rightarrow|f(x)| \quad$ (ii) $f(x) \rightarrow f(|x|) \mid$

For Example:

              y = f(x)                                                y = |f(x)|                                             y = f(|x|)

              y = |f(|x|)|

$\begin{aligned} & y=f(x) \rightarrow|y|=f(x) \\ & y=f(x) \text { is given }\end{aligned}$

Remove the part of the graph which lies below X-axis

Plot the remaining part 

take the mirror image of the portion that lies above the x-axis about the x-axis.