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Chapter 8: Problem 5

Describe each function as a transformation of the graph of the parent function \(y=\frac{1}{x}\). Then sketch a graph of each function and list values that are not part of the domain. a. \(y=\frac{1}{x}\) b. \(y=\frac{1}{x-5}-2\) c. \(y=\frac{0.5}{x}+3\) d. \(y=\frac{-3}{x+3}\)

Short Answer

Expert verified
For each function: - a: Same as parent, domain \( x \neq 0 \). - b: Shift right 5, down 2; domain \( x \neq 5 \). - c: Vertical compression, up 3; domain \( x \neq 0 \). - d: Left 3, reflect, stretch 3; domain \( x \neq -3 \).

Step by step solution

01

Understanding the Parent Function

The parent function is given by \( y = \frac{1}{x} \). This is a basic rational function defined everywhere except at \( x = 0 \) where it is undefined because division by zero is not possible. The graph is a hyperbola with two branches, one in the first quadrant and one in the third quadrant.
02

Analyzing Function a

The function in part (a) is \( y = \frac{1}{x} \). Since this is identical to the parent function, there are no transformations. The domain is all real numbers except \( x = 0 \).
03

Transformations in Function b

The function \( y = \frac{1}{x-5} - 2 \) represents two transformations:1. Horizontal shift: The \( x \) is replaced by \( x-5 \), indicating a horizontal shift 5 units to the right.2. Vertical shift: The '-2' indicates a vertical shift 2 units down.The domain is all real numbers except \( x = 5 \).
04

Transformations in Function c

The function \( y = \frac{0.5}{x} + 3 \) has two transformations:1. Vertical stretch: The numerator is 0.5, which compresses the graph vertically compared to \( y = \frac{1}{x} \).2. Vertical shift: The '+3' indicates a vertical shift 3 units up.The domain is all real numbers except \( x = 0 \).
05

Transformations in Function d

The function \( y = \frac{-3}{x+3} \) includes:1. Horizontal shift: \( x \) is replaced by \( x+3 \), indicating a shift 3 units to the left.2. Reflection and vertical stretch: The '-3' in the numerator indicates a reflection across the x-axis and vertical stretching by a factor of 3.The domain is all real numbers except \( x = -3 \).

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Key Concepts

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

Parent Function
The parent function for many rational functions is the simple equation \( y = \frac{1}{x} \). This function is one of the most fundamental forms in mathematics known as a rational function. It is defined at all points except where the denominator equals zero, making the domain all real numbers except for \( x = 0 \). The graph of this function is characterized by a hyperbola with two curves, one in the first quadrant and one in the third quadrant. These curves are mirror images of each other.

  • The key feature of the parent function is its vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \).
  • The behavior of the function approaches infinity as \( x \) approaches zero.
Understanding this basic graph will assist you in visualizing and predicting how transformations impact the graph.
Transformations
Transformations modify the graph of the parent function \( y = \frac{1}{x} \) in various ways, including shifting, stretching, compressing, and reflecting. These transformations alter the position or appearance of the graph without changing the fundamental nature of the rational function.
  • Horizontal Shift: Replacing \( x \) with \( x - h \) (or \( x + h \)) moves the graph horizontally by \( h \) units. For example, \( y = \frac{1}{x-5} \) is shifted 5 units to the right.
  • Vertical Shift: Adding or subtracting a value \( k \) to the function moves the graph up or down. For instance, \( y = \frac{1}{x-5} - 2 \) is shifted down by 2 units.
  • Reflection: Multiplying by a negative value \( -a \) reflects the graph across an axis. \( y = \frac{-3}{x+3} \) is reflected across the x-axis.
  • Vertical Stretch or Compression: Changing the coefficient in the numerator stretches or compresses the graph vertically. A numerator greater than 1 stretches the graph, while less than 1 compresses it. For instance, \( y = \frac{0.5}{x} \) compresses the graph.
These transformations help in identifying how each change affects the graph's shape and position.
Domain
The domain of a rational function is crucial because it tells us the values of \( x \) where the function is defined. For the parent function \( y = \frac{1}{x} \), the domain is all real numbers except \( x = 0 \), because division by zero is undefined.

Each transformation can affect the domain, mainly due to horizontal shifts.
  • In \( y = \frac{1}{x-5} - 2 \), the horizontal shift of 5 units makes \( x = 5 \) the point where the function is undefined.
  • In \( y = \frac{0.5}{x} + 3 \), the domain remains all real numbers except \( x = 0 \).
  • For \( y = \frac{-3}{x+3} \), the shift makes \( x = -3 \) the excluded value from the domain.
Remember, each transformation that changes the denominator's form directly impacts the domain, determining where the graph has vertical asymptotes.
Graphing
Graphing a rational function requires knowledge of its parent function and how transformations alter it. Begin by sketching the basic shape of \( y = \frac{1}{x} \), a hyperbola, and apply the transformations systematically.
  • Identify vertical and horizontal asymptotes dictated by the domain and any vertical and horizontal shifts.
  • Consider any reflections and adjust the graph accordingly. A negative coefficient in the numerator, like in \( y = \frac{-3}{x+3} \), indicates a reflection across the x-axis.
  • When vertical stretching or compressing is involved, modify the curvature's steepness appropriately.
By applying transformations one by one, you gain a clearer picture of the final graph. It's crucial to check how the asymptotes and key points have moved based on the function's new form. This process helps in sketching an accurate graph.

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Most popular questions from this chapter

Tacoma and Jared are doing a "walker" investigation. Tacoma starts \(2 \mathrm{~m}\) from the motion sensor. He walks away at a rate of \(0.5 \mathrm{~m} / \mathrm{s}\) for \(6 \mathrm{~s}\). Then he walks back toward the sensor at a rate of \(0.5 \mathrm{~m} / \mathrm{s}\) for \(3 \mathrm{~s}\). a. Sketch a time-distance graph for Tacoma's walk. b. Write an equation that fits the graph.

APPLICATION In 2004 , the world population was estimated to be \(6.4\) billion, with an annual growth rate of \(1.14 \%\). (Central Intelligence Agency, www.cia.gov) a. Define input and output variables for this situation. b. Without finding an equation, sketch a graph of this situation for 1995 to 2015 . c. What one point on the graph do you know for sure? d. Write a function that models this situation. Graph your function on your calculator and name an appropriate window. e. Use your graph to estimate the population to the nearest tenth of a billion in 1995 and 2015 . (Assume a constant growth rate during this period.)

Use the properties of exponents to rewrite each expression without negative exponents. a. \(\left(2^{3}\right)^{-3}\) (a) b. \(\left(5^{2}\right)^{5}\) c. \(\left(2^{4} \cdot 3^{2}\right)^{3}\) d. \(\left(3^{2} \cdot x^{3}\right)^{-4}\)

Solve each system of equations. a. \(\left\\{\begin{array}{l}y=5+2 x \\ y=8-2 x\end{array}\right.\) b. \(\left\\{\begin{array}{l}y=-2+3(x-4) \\ y=3+5(x-2)\end{array}\right.\) c. \(\left\\{\begin{array}{l}2 x+7 y=13 \\ 5 x-14 y=1\end{array}\right.\)

Solve each equation. a. \(5=-3+2 x\) b. \(-4=-8+3(x-2)\) c. \(7+2 x=3+x\)
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