Question

i am so lost. can someone help?? the question is in the picture.

Answer 1

We are given the following function:

$f(x)=x^2$

We are asked to do the following tranformations:

1. Shift down 5 units and left 77 units.

First, to shift a function down a number "n" of units we follow the next rule:

$h(x)=f(x)-n$

And, to shift a function "m" units to the left we use the following rule:

$h(x)=f(x+m)$

Applying both rules simultaneously we get:

$h(x)=(x+77)^2-5$

2. Reflect about the y-axis.

The rule to reflect a function about the y-axis is the following:

$r(x)=h(-x)$

The means that we will change "x" for "-x" in the function we are going to reflect. Applying the rule we get:

$r(x)=(-x+77)^2-5$

3. Compress vertically by a factor of 2.

To compress a function vertically by a factor "m" we multiply the entire function by 1/m, like this:

$g(x)=\frac{1}{m}r(x)$

Applying the rule we get:

$g(x)=\frac{1}{2}((-x+77)^2-5)$

And thus we get to the final function.