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(b) Describe fully a sequence of transformations which transform the graph of
y=(2)/(x^(2)-1) to the graph of
y=6+(4)/(4x^(2)-1)

(b) Describe fully a sequence of transformations which transform the graph of $y=\frac{2}{x^{2}-1}$ to the graph of $y=6+\frac{4}{4 x^{2}-1}$

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Find the vertex of the transformed function

Full solution

Q. (b) Describe fully a sequence of transformations which transform the graph of

y=\frac{2}{x^{2}-1}

to the graph of

y=6+\frac{4}{4 x^{2}-1}

Analyze original function: We start by analyzing the original function $y = \frac{2}{x^2 - 1}$ .
Vertical shift and addition: The first transformation we notice is the addition of $6$ to the entire function. This is a vertical shift upwards by $6$ units. $\newline$ $y = \frac{2}{x^2 - 1}$ becomes $y = 6 + \frac{2}{x^2 - 1}$ .
Vertical stretch by factor: Next, we observe that the numerator of the fraction has changed from $2$ to $4$ . This is a vertical stretch by a factor of $2$ . $\newline$ $y = 6 + \frac{2}{x^2 - 1}$ becomes $y = 6 + \frac{4}{x^2 - 1}$ .
Horizontal compression by factor: Then, we see that the denominator has changed from $(x^2 - 1)$ to $(4x^2 - 1)$ . This is a horizontal compression by a factor of $\frac{1}{2}$ , since the $x^2$ term is multiplied by $4$ . $y = 6 + \frac{4}{x^2 - 1}$ becomes $y = 6 + \frac{4}{4x^2 - 1}$ .
Check for horizontal or vertical shift: Finally, we need to check if there is any horizontal or vertical shift associated with the change in the denominator from $(x^2 - 1)$ to $(4x^2 - 1)$ . Since the constant term in the denominator remains $-1$ , there is no horizontal or vertical shift associated with this change.

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