Question

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{9 x-3}{x+7} $$

Short answer

The function \(g(x)=\frac{9 x-3}{x+7}\) can be rewritten as \(g(x)=9-\frac{129}{x}\). After graphing this function, it can be seen that it has a horizontal asymptote at \(y = 9,\) a vertical asymptote at \(x =0\), and a hole at \(x =0\). The graph of \(g(x)\) is a transformation of \(f(x) = \frac{1}{x}\) with a vertical shift of 9 units up and a horizontal compression by a factor of 129.

Step-by-step solution

Rewriting the Function

To begin with, the function is rewritten as \(g(x)=\frac{9}{1+\frac{7}{x}}\). This looks more like the required format now. But, it should be noted that \(9 = 9(1-\frac{3}{9x})\) and \(1+\frac{7}{x} = (1-\frac{7}{x})+\frac{14}{x}\). Therefore, the function is now \(g(x)=\frac{9(1-\frac{3}{9x})}{(1-\frac{7}{x})+\frac{14}{x}}\). Simplifying the terms, it can be written as \(g(x)=9(1-\frac{3}{9x})-\frac{126}{x}\), which is equal to \(g(x)=9-\frac{3}{x}-\frac{126}{x}\). Now, combining the fractions with \(x\) as the denominator, the function is represented as \(g(x)=9-\frac{129}{x}\).

Graphing the Function

To illustrate graph \(g(x)=9-\frac{129}{x}\), a rational function graph is created. The horizontal asymptote is \(y =9 \) and the vertical asymptote is \(x=0\). It crosses the y-axis at \(y = -129 \). Because x cannot be 0, the graph has a hole at \(x = 0\).

Describing the Transformation

To start with, the graph of \(g(x)\) is a transformation of the graph of \(f(x)=\frac{1}{x}\). There is a vertical shift of 9 units up because of the \(+9\) in \(g(x)=9-\frac{129}{x}\). There is also a horizontal compression by a factor of 129 which results in the function approaching the asymptotes at a quicker pace.

Key concepts

Rational Function Transformations

Understanding rational function transformations involves studying changes to the graph of a basic rational function. In these transformations, we shift, stretch, or compress the graph.

For the given function, the transformation involves a vertical shift of 9 units upward due to the '+9' in the equation. This '+9' moves the entire graph up on the y-axis, changing its position compared to the original function, which is represented by the parent function, in this case, \(f(x)=\frac{1}{x}\).

Additionally, there's a horizontal compression by a factor of 129. Imagine the x-values being pulled closer together, which makes the graph of \(g(x)\) appear more 'steep' as it approaches the asymptotes. It's crucial to note that these transformations do not affect the asymptotes themselves but rather how the graph approaches them.

Horizontal Asymptote

A horizontal asymptote of a rational function is a horizontal line that the graph approaches as the x-values head towards positive or negative infinity. It's akin to a guide that tells us where the function is heading in the long run, but it never quite gets there.

For our function \(g(x) = 9 - \frac{129}{x}\), the horizontal asymptote is \(y = 9\). This is because the x-value in the denominator has less influence as x grows larger or more negative. Essentially, the \(\frac{129}{x}\) part becomes insignificant compared to the constant 9, and the graph will level off towards this line as it stretches out infinitely to the left or right.

Vertical Asymptote

The vertical asymptote is related to values that x cannot be. It's a vertical line at some x-value where the function tends towards infinity or negative infinity as it gets really close to that x-value.

In the context of our function \(g(x) = 9 - \frac{129}{x}\), the vertical asymptote is \(x = 0\). This is because the denominator of the rational expression cannot be zero (as division by zero is undefined), and as x gets extremely close to zero, the term \(-\frac{129}{x}\) grows without bound. The graph will get closer and closer to the line \(x = 0\), but it will never touch or cross it.

Rewriting Functions

Rewriting functions is a process of rearranging the function into a different form that reveals certain characteristics, like asymptotes and transformations, more clearly. This practice is invaluable in understanding and graphing the behavior of rational functions.

For instance, in our solution, we converted \(g(x) = \frac{9x-3}{x+7}\) into \(g(x) = 9 - \frac{129}{x}\) by factoring and simplifying. This new form highlights the vertical shift, indicated by the constant term, and the horizontal compression, shown by the coefficient in the denominator. By deconstructing and rewriting the function into this transformed state, we can immediately discern its graphical characteristics and asymptotic behavior.