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Chapter 11: Problem 7

Use reflections and/or translations to graph each rational function. $$f(x)=-\frac{3}{x}$$

Short Answer

Expert verified
Reflect \(\frac{1}{x}\) across the x-axis to get \ -\frac{1}{x} \ and then vertically stretch it by a factor of 3 to get \ -\frac{3}{x}\.

Step by step solution

01

- Identify the parent function

The parent function for this rational function is \(g(x) = \frac{1}{x}\).
02

- Apply the reflection

The negative sign in front of the fraction indicates a reflection across the x-axis. Reflecting \(g(x)\) across the x-axis, we get \(h(x) = -\frac{1}{x}\).
03

- Apply the vertical stretch

The coefficient \(-3\) outside the fraction indicates a vertical stretch by a factor of 3. Thus, applying this transformation to \(h(x)\), we get \(f(x) = -\frac{3}{x}\).
04

- Graph the function

Graph the final function \(f(x) = -\frac{3}{x}\). Plot key points and show the asymptotes. The vertical and horizontal asymptotes are at x = 0 and y = 0, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection Across X-Axis
When graphing rational functions, understanding reflections is key. In our exercise, we started with the parent function, which is \(\frac{1}{x}\).
The negative sign in front of the function indicates a reflection across the x-axis. This means every point on the graph of \( \frac{1}{x} \) will be reflected, or flipped, over the x-axis.
This transformation changes the graph from the first and third quadrants to the second and fourth quadrants. So, the function \( \frac{1}{x} \) transforms into \( -\frac{1}{x} \).
This step is essential to correctly interpret both the shape and position of the rational function.
Vertical Stretch
After reflecting a function across the x-axis, we often need to apply vertical stretches or compressions.
In our case, the reflection step yielded \( -\frac{1}{x} \). To apply a vertical stretch by a factor of 3, you multiply the whole function by 3.
This makes our function \( -\frac{3}{x} \).
Stretching vertically means that the graph will move further away from the x-axis if the factor is greater than 1.
So, every y-value of the function will be 3 times farther from the x-axis compared to \( -\frac{1}{x} \). Visually, this transformation makes the graph steeper.
This helps depict how much quicker the values of the function change as x moves away from zero.
Asymptotes
Asymptotes are lines that the graph of a function approaches but never actually touches.
For the rational function \( f(x) = -\frac{3}{x} \), there are two important asymptotes to consider: a vertical asymptote and a horizontal asymptote.
The vertical asymptote occurs where the denominator equals zero. For our function, this is at \( x = 0 \).
This means that as x approaches zero from the left or the right, the function's value tends to infinity or negative infinity. Next, the horizontal asymptote indicates the value that the function approaches as x moves towards positive or negative infinity.
For our function, this is y = 0. This is because, as x becomes very large or very small, the value of \( -\frac{3}{x} \) approaches zero but never actually reaches it.
These asymptotes are crucial for understanding the behavior of the graph and are always a significant aspect to plot and consider.

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Most popular questions from this chapter

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