Chapter 3.5, Problem 63PFA

To determine

To find: Which function represents a vertical stretch of the graph $f\left(x\right)=x$

Answer to Problem 63PFA

  $g\left(x\right)=7x$

Explanation of Solution

Given information:

  $f\left(x\right)=x$

Concept:

A Translation is a movement of the graph either horizontally parallel to x-axis or vertically parallel to y-axis.

The graph of $y=f\left(x\right)$ where $f\left(x\right)=x$ is same as the graph of $y=x$ . Writing the equations as functions in the form of $f\left(x\right)$ is useful when applying translations and reflections to graphs.

If $g\left(x\right)=f\left(x\right)+k$ it is known as vertical shift.

If $k>0$ it moves upwards.

If $k<0$ it moves downwards.

If $g\left(x\right)=f\left(x+h\right)$ it is known as horizontal shift.

If $h<0$ moves it right.

If $h>0$ moves it left.

If $g\left(x\right)=-f\left(x\right)$ it is known as Vertical reflection it means reflects about X-axis

If $g\left(x\right)=f\left(-x\right)$ it is known as Horizontal reflection it means reflects about Y-axis

If $g\left(x\right)=af\left(x\right)$

If $a>1$ it is known as Vertical stretch.

If $0<a<1$ it is known as Vertical compression.

If $g\left(x\right)=f\left(bx\right)$

If $b>1$ it is known as Horizontal compression.

If $0<b<1$ it is known as Horizontal Stretch.

Calculation:

In option A given

  $g\left(x\right)=x+7$

  $g\left(x\right)=f\left(x\right)+7$

If $g\left(x\right)=f\left(x\right)+k$ it is known as vertical shift.

If $k>0$ it moves upwards.

If $k<0$ it moves downwards.

Here $k=7$ as $7>0$ there will be vertical shift of 7 units upwards.

In option B given

  $g\left(x\right)=\left(0.7x\right)$

  $\Rightarrow g\left(x\right)=f\left(0.7x\right)$

If $g\left(x\right)=f\left(bx\right)$

If $b>1$ it is known as Horizontal compression.

If $0<b<1$ it is known as Horizontal Stretch.

Here $b=0.7$ as $0<0.7<1$ so, there will be Horizontal stretch.

In Option C given

  $g\left(x\right)=\frac{1}{7}x$

  $\Rightarrow g\left(x\right)=\frac{1}{7}f\left(x\right)$

If $g\left(x\right)=af\left(x\right)$

If $a>1$ it is known as Vertical stretch.

If $0<a<1$ it is known as Vertical compression.

Here $a=\frac{1}{7}$ as $0<\frac{1}{7}<1$ so, there will be Vertical Compression.

In Option D given

  $g\left(x\right)=7x$

  $\Rightarrow g\left(x\right)=7f\left(x\right)$

If $g\left(x\right)=af\left(x\right)$

If $a>1$ it is known as Vertical stretch.

If $0<a<1$ it is known as Vertical compression.

Here $a=7$ as $7>1$ so, there will be Vertical Stretch.

Therefore, there the function which represents Vertical stretch of the graph $f\left(x\right)=x$ is Option D, $g\left(x\right)=7x$