Rational function transformation

[woof123]

the question is: Rewrite the rational equation y=(-5x-18)/(x+4) to show how it is a transformation of y=1/x. describe transformations

looks like it is shifted 4 to left, then stretched by factor of -5x-18. Is that accurate? would you elaborate beyond that?

 

[MarkFL]

We are given:

\(\displaystyle y=-\frac{5x+18}{x+4}\)

And to show this is a transformation of:

\(\displaystyle f(x)=\frac{1}{x}\)

We need to write $y$ in the form:

\(\displaystyle y(x)=a\frac{1}{x-b}+c\)

where $a$ is the vertical stretching factor, $b$ is the horizontal shift, and $c$ is the vertical shift.

So, I would observe that:

\(\displaystyle 5x+18=5(x+4)-2\)

Can you continue?

 

[woof123]

sorry but I don't see how factoring the numberator helps get it into the form you described.[desmos="-10,10,-10,10"]y=-(5x+18)/(x+4)[/desmos] I see the function is shifted left 4 (as accomplished by the denominator but I'm not understanding how the numerator gets it shifted down 4 (approx). Is the numberator you have now factored considered "c"?

 

[topsquark]

What MarkFL is saying is that you need to change the form of your function. So do some division:
\(\displaystyle y = - \frac{5x + 18}{x + 4} = - \left ( 5 + \frac{-2}{x + 4} \right )\)

How do you compare this with \(\displaystyle a \frac{1}{x - b} + c\)?

-Dan

 

[woof123]

sorry but i don't understand how those two expressions are equal. what happened to the 18

 

[topsquark]

sorry but i don't understand how those two expressions are equal. what happened to the 18

I'm not quite good enough with tables to show you the division, so I'll prove it backward:
\(\displaystyle 5 + \frac{-2}{x + 4} = \frac{5(x + 4)}{x + 4} + \frac{-2}{x + 4}\)

\(\displaystyle = \frac{5(x + 4) - 2}{x + 4} = \frac{5x + 20 - 2}{x + 4} = \frac{5x + 18}{x + 4}\)

This site has a worked example. You should have done this in either the class you are taking now or previously.

-Dan

 

[MarkFL]

This is what I intended for you to do:

\(\displaystyle y=-\frac{5x+18}{x+4}=-\frac{5(x+4)-2}{x+4}=\frac{2}{x+4}-5\)

And so we see that the vertical stretch is 2, the horizontal shift is 4 units to the left, and the vertical shift is 5 units down.

 

[suluclac]

Just do the long division. After that, you should arrive with an equation that allows division by x + 4.
Such equation should be something like -(5x + 18) = . . ..