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 \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{7 x-20}{x+13} $$

Short answer

The function \(g(x)=\frac{7x - 20}{x + 13}\) rewrites to \(g(x) = -\frac{286}{x + 13} + 7\), and is a transformation of \(f(x) = \frac{a}{x}\). The transformation in this case includes a right shift by 13 units, a stretching by a factor of -286 and vertical shift upwards by 7 units.

Step-by-step solution

Decompose the given function

To rewrite \(g(x)=\frac{7 x - 20}{x + 13}\) in a form of \(g(x) = \frac{a}{x - h} + k\), first we decompose the function into two simpler fractions that we could then adjust for the required form.

Simplify the decomposed function

We simplify the decomposed function to \(g(x) = 7 - \frac{286}{x + 13}\). This is done by arranging 7x and 20 on their own divided by x + 13, and then simplifying both of them.

Adjust to the desired form

Our new function is \(g(x) = 7 - \frac{286}{x + 13}, \) but our goal is to adjust it to the form \(g(x)\) = a/(x - H) + K, which means rewriting the function again to \(g(x) = -\frac{286}{x + 13} + 7\). Here, the transformation parameters are h = -13, a = -286, and k = 7.

Describe the transformations

The function \(g(x) = \frac{a}{x - h} + k\), shows a horizontal shift (h), a vertical stretch by factor of a, and a vertical shift(k). Either one of these transformation changes the graph of the function. So, \(g(x) = -\frac{286}{x + 13} + 7\) shows a right shift by 13 units, a stretching by a factor of -286 and vertical shift upwards by 7 units compared to the graph of the function \(f(x) = \frac{a}{x}\).

Key concepts

Horizontal Shift

When we talk about the transformation of functions, a horizontal shift is one of the basic maneuvers you'll need to understand.

Imagine sliding a function to the left or right on the coordinate plane without altering its shape. That's exactly what a horizontal shift does. In mathematical terms, if you have a function like \( f(x) = \frac{a}{x} \), and you want to shift it horizontally, the new form would be \( g(x) = \frac{a}{x - h} \).

The variable 'h' represents the number of units you'll move the graph. If 'h' is positive, the function slides to the right. If it's negative, it goes to the left. For instance, in our exercise, the function \( g(x) = -\frac{286}{x + 13} + 7 \) is obtained by shifting the original function \( f(x) = \frac{a}{x} \) to the right by 13 units, as indicated by the '+13' in the denominator.

Vertical Stretch

What Happens When a Graph Stretches Vertically?

Picture a rubber band being pulled apart vertically. A vertical stretch of a graph feels somewhat similar. It's a transformation that distorts the function by pulling it away from the x-axis if its factor is greater than 1, or compressing it towards the x-axis if the factor is between 0 and 1.

In the original function \( f(x) = \frac{a}{x} \), 'a' is the factor that determines the intensity of the stretch. Revisiting our exercise, the factor 'a' in \( g(x) = -\frac{286}{x + 13} + 7 \) is -286. This indicates a substantial vertical stretch combined with a reflection across the x-axis due to the negative sign. Essentially, the graph becomes taller and more narrow, while also being flipped upside down.

Rational Functions

Delving into rational functions opens a door to equations involving ratios of polynomials.

A classic example is \( f(x) = \frac{1}{x} \), which is a simple rational function showcasing hyperbolic behavior. But what makes rational functions particularly interesting is their asymptotic nature; they approach certain lines (asymptotes) without ever touching them.

The function \( g(x) = \frac{7x - 20}{x + 13} \) from our exercise is a rational function. Such functions often require transformations to reveal underlying shifts and stretches. Understanding how to decompose and rewrite them, as we did in steps 1 and 2, is essential for identifying the shifts from the original function.

Graphing Transformations

The art of graphing transformations is fundamental in visualizing how a function has been altered.

After applying shifts and stretches to a function, its graph will look different. It's crucial to recognize that transformations follow a certain order: stretches or compressions come first, followed by shifts. When graphing, always pay attention to these changes to accurately depict the new function.

In the final step of our solved problem, describing the transformation includes acknowledging that the function has been shifted horizontally to the right by 13 units and vertically upward by 7 units. Moreover, the negative stretch factor indicates that the graph is not only stretched but also reflected. Plotting each part carefully will guide us to the accurate representation of the transformed function.