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

Here, we're asked to graph the rational function using transformations. And here, I have the function g of x is equal to negative one over x plus 3, which I know I'm just going to be transforming the function 1 over x. So starting with step number 1 here, we want to go ahead and plot our vertical asymptote at x equals h. Now here, I don't have any horizontal shift, so h is simply 0. So my vertical asymptote goes right at that axis at x equals 0. Now, my horizontal axis or my horizontal asymptote will be at y equals to k. Here k is 3, so I need to go ahead and plot that horizontal asymptote right up here at y equals 3. Now for step 3, we want to determine if there is a reflection happening, which there is because I have this negative right outside here. So, yes, there is a reflection. I need to go ahead and reflect these points over the correct axis, in this case, the x axis, but it doesn't really matter because it's an odd function. It will end up being the same either way. So let's go ahead and reflect these two points here. Now, reflecting these points remember is the same as if I fold it at the axis and just kind of stamp those points on the other side. So here is the position of my new points. But now that I've reflected them, I also need to shift them according to what I have for h and k. And here h is 0 k is 3. So, I need to go ahead and shift these 3 units up. So shifting this first point I'm going to end up right here and then shifting my other point 3 units up I'm going to end up right here. So I have everything that I need for my new graph and I can go ahead and end up with my final step which is to sketch my curves approaching the asymptotes. So sketching my curves here, I know we're going to be approaching that horizontal asymptote here, the vertical asymptote here, and the same happening from this point. So we see we have the exact same shape, really similar looking function, it's just been formed from 1 over x. So let's go ahead and identify our domain and range. Now remember, our domain is always going to go from negative infinity until we reach that asymptote. So in this case it will be at 0 and then it will pick up on the other side continuing on from that asymptote to infinity, of course, enclosed in parentheses and not square brackets because those values are not included. Then the same thing for the range here from negative infinity to that asymptote, so in this case from negative infinity to 3, and then from 3 to infinity. So that's all for this problem we've completed our graph I'll see you in the next video.

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.
$f\left(x\right)=-\frac{1}{x}+3$

Verified step by step guidance

Start with the basic rational function $ f(x) = \frac{1}{x} $. This function has a vertical asymptote at $ x = 0 $ and a horizontal asymptote at $ y = 0 $.

Apply the transformation for the negative sign: $ f(x) = -\frac{1}{x} $. This reflects the graph across the x-axis, so the graph will be in the second and fourth quadrants.

Next, apply the vertical shift: $ f(x) = -\frac{1}{x} + 3 $. This shifts the entire graph up by 3 units. The horizontal asymptote moves from $ y = 0 $ to $ y = 3 $.

Identify the new vertical asymptote, which remains at $ x = 0 $ since there is no horizontal shift.

Plot the transformed graph, ensuring the curve approaches the asymptotes: the graph should approach $ y = 3 $ as $ x \to \pm\infty $ and approach the vertical asymptote at $ x = 0 $ from both sides.