Graph the rational function using transformations. f ( x ) = 1 ( x + 3 ) 2 − 2 f\left(x\right)=\frac{1}{\left(x+3\right)^2}-2 f ( x ) = ( x + 3 ) 2 1 − 2

In this problem, we're asked to graph the rational function using transformations, and here I have the function g of x is equal to 1 over x plus 3 squared minus 2. So I'll be transforming the function 1 over x squared. Now starting with step 1 here, we want to go ahead and plot our vertical asymptote at x equals h. And here I have x plus 3, so we need to be careful with the sign here. Remember, when we're dealing with transformations, it's always x minus h. So if I have a plus here, that tells me that this is x minus negative 3. That double negative gives me that positive. So this vertical asymptote will be at x equals negative 3, taking that sign into account. So plotting that on my graph here using a dash line, now I can move on to my horizontal asymptote at y equals k. Now y equals k is at negative 2 here, so I'm gonna go ahead and plot that using a dash line at y equals negative 2. Now moving on to step 3, we need to determine if there is a reflection and there is not a reflection. I do not see any extra negatives there, so I don't have to worry about that. But moving on to 3 b, we do need to go ahead and shift those test points by h comma k, in this case, by negative 3 and negative 2. So let's go ahead and do that. Now with my points here, I wanna move them 3 units to the left. So starting with this point, 3 units to the left and then 2 units down. So right here is gonna be my first point. And then my second point, same thing, 3 units to the left and then 2 units down. So 3 left, 1, 2, 3, and 2 down. So I'm going to end up right here. Now that we have shifted our points we can go ahead and move on to our final step which is going to be to sketch our curves approaching our asymptotes. So going back to our graph over here I can go ahead and sketch those curves. No reflection happens so we're going to have the exact same shape in the exact same reflection happens so we're going to have the exact same shape in the exact same direction and here is my new graph. Now let's go ahead and identify our domain and our range here. Remember, our domain will always go from negative infinity to the asymptote so in this case our domain will always go from negative infinity to that asymptote. So in this case, from negative infinity to negative 3, and then from negative 3 to positive infinity. And then finally our range here since we are only in that part above our vertical or above our horizontal asymptote this is going to go from my horizontal asymptote to infinity. So it goes up to infinity, but from my horizontal asymptote at negative 2, so from negative 2 to infinity. Now that we've identified that domain in our range we are completely done here, thanks for watching and I'll see you in the next one.

5. Rational Functions

Graphing Rational Functions

Precalculus

5. Rational Functions

Graphing Rational Functions

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Multiple Choice

Graph the rational function using transformations.
$f\left(x\right)=\frac{1}{\left(x+3\right)^2}-2$

Verified step by step guidance

Start with the basic rational function f(x) = 1/x^2, which has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

Apply a horizontal shift to the function by replacing x with (x + 3). This shifts the graph 3 units to the left, moving the vertical asymptote to x = -3.

Next, apply a vertical shift by subtracting 2 from the function. This moves the entire graph down by 2 units, changing the horizontal asymptote from y = 0 to y = -2.

The transformed function is f(x) = 1/(x + 3)^2 - 2. The graph now has a vertical asymptote at x = -3 and a horizontal asymptote at y = -2.

Sketch the graph using the asymptotes as guides. The graph will approach the vertical asymptote at x = -3 and the horizontal asymptote at y = -2, with the curve opening upwards on both sides of the vertical asymptote.