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# 4) Let f(x) = 1/ x be the parent function. Let g(x) = (3 x - 10)/( x - 4) be a transformation of f(x) .

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4) Let f(x) = 1/x be the parent function. Let g(x) = (3x - 10)/(x - 4) be a transformation of f(x).

(a) Rewrite g(x) in the form a + (k/(x - 4)).

(b) Describe in words the transformations that take f(x) to g(x).

(c) If f(x) contains the points (-2, -½) and (1, 1), find the corresponding coordinates on g(x) using the transformation rules from part (b).

(d) Find the equations of the vertical and horizontal asymptotes.

Mar 20, 2020

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(a) Rewrite g(x) in the form a+(k/(x-4))

I suppose that meant to simplify the division $$\frac{(3x-10)}{x-4}$$

$$Thus,3+\frac{2}{x-4}$$ Same as the form.

Mar 20, 2020
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Thanks guest, you made a good start.

4) Let f(x) = 1/x be the parent function. Let g(x) = (3x - 10)/(x - 4) be a transformation of f(x).

a)

$$f(x)=\frac{1}{x}\qquad g(x)=\frac{3x-10}{x-4}\\~\\ g(x)=\frac{3x-10}{x-4}\\ g(x)=\frac{3x-12+2}{x-4}\\ g(x)=\frac{3(x-4)+2}{x-4}\\ g(x)=3+\frac{2}{x-4}\\$$

Now let's look at this transformation

$$y=\frac{1}{x}\\ \text{If I translate the graph right (positive x direction) by 4 units I get }\\ y=\frac{1}{x+4}\\~\\ \text{If then translate the graph up 3 units i get }\\ y-3=\frac{1}{x+4}\\ y=3+\frac{1}{x+4}\\$$

(b) Describe in words the transformations that take f(x) to g(x).

So to transform f(x) to g(x) I must translate every point 4 units to the right and 3 units up.

(c) If f(x) contains the points (-2, -½) and (1, 1), find the corresponding coordinates on g(x) using the transformation rules from part (b).

$$(-2,-0.5)\rightarrow (-2+4,-0.5+3)=(2,2.5)$$

Here is the graph:

https://www.desmos.com/calculator/gyqdgjfupu

(d) Find the equations of the vertical and horizontal asymptotes.

You should be able to finish the rest.

Mar 20, 2020