Chapter 13, Problem 22MCQ

Solution

i.

To hypothesize: how the formula needs to be changed to draw a sine curve having midline above or below $x-\text{axis}$ .

  $y=a\sin b\theta +k$ , $y=a\sin b\theta -k$

Given:

The functions is $y=a\sin b\theta$ .

Calculation:

The curve of the sine function is symmetric to the origin. So, the midline of the sine curve is $x-\text{axis}$ . To make the midline above $x-\text{axis}$ , one has to add the number of units at which midline must be lies.

For example, if the midline must be lies $k$ unit above $x-\text{axis}$

then the it can be modified as follows:

  $y=a\sin b\theta +k$

Similarly, if midline lies $k$ unit below $x-\text{axis}$ then the formula can be modified as $y=a\sin b\theta -k$ .

Conclusion:

Hence, making the midline $k$ unit above modify the function as $y=a\sin b\theta +k$ . Similarly, to draw the midline of sine curve $k$ unit below modify the function as $y=a\sin b\theta -k$ .

ii.

To describe: how to translate function left or right.

Right translation : $y=a\sin \left(b-k\right)\theta$

Left translation: $y=a\sin \left(b+k\right)\theta$

Given:

The functions is $y=a\sin b\theta$ .

Concept:

If the function is $f\left(x\right)$ then to translate it $h$ unit right, write the function as $g\left(x\right)=f\left(x-h\right)$ and to translate it $h$ unit left write the function as $g\left(x\right)=f\left(x+h\right)$ .

Calculation:

To translate the function $y=a\sin b\theta$ , $k$

unit right rewrite the function as follows:

  $y=a\sin \left(b-k\right)\theta$

And the function can be translated $k$ unit left by adding $k$ to the $b$ .

  $y=a\sin \left(b+k\right)\theta$

Conclusion:

Hence, $y=a\sin \left(b-k\right)\theta$ for right translation of $k$ unit and $y=a\sin \left(b+k\right)\theta$ for left translation of $k$ unit.