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Chapter 4: Problem 20

Graph the rational function by applying transformations to the graph of \(y=\frac{1}{x}\). $$j(x)=\frac{3 x-7}{x-2}$$

Short Answer

Expert verified
The function \( j(x) = \frac{3x - 7}{x - 2} \) has a vertical asymptote at \( x = 2 \) and a horizontal asymptote at \( y = 3 \). Shift the graph of \( y = \frac{1}{x} \) right by 2 units and up by 3 units.

Step by step solution

01

Identify the Parent Function

The parent function is given as \( y = \frac{1}{x} \). This function is a hyperbola with vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \) respectively.
02

Rewrite in the Transformative Form

To transform \( j(x) = \frac{3x - 7}{x - 2} \), consider expressing it like \( y = \frac{1}{x} \) by rewriting it as \( j(x) = \frac{3(x - \frac{7}{3})}{x - 2} \).
03

Determine Asymptotes

The vertical asymptote is found by setting the denominator equal to zero: \( x - 2 = 0 \Rightarrow x = 2 \). The horizontal asymptote is determined by the coefficients in the highest degree terms: \( \frac{3}{1} = 3 \), so \( y = 3 \) is the horizontal asymptote.
04

Identify the Transformations

The horizontal shift is determined by the modification in the denominator, resulting in a shift to the right by 2 units, from \(x = 0\) to \(x = 2\). The horizontal asymptote \(y = 0\) shifts to \(y = 3\) indicating vertical translation.
05

Graphing the Function

Draw the asymptotes on the graph. The graph of \( y = \frac{1}{x} \) will be shifted right 2 units and moved up 3 units. Plot key points to reflect these transformations, maintaining the hyperbolic shape.

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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 of the given rational function is always a great starting point for analysis. Here, we have the parent function defined as \( y = \frac{1}{x} \). This is a simple hyperbola and a crucial reference for all foundational transformations. It's characterized by symmetry in its quadrants and has asymptotes at both the vertical and horizontal axes:
  • Vertical asymptote: \( x = 0 \)
  • Horizontal asymptote: \( y = 0 \)
The graph of this parent function is sharply curved, with two disconnected branches. Each branch approaches the asymptotes but never actually touches them. Starting with this parent function allows us to predict and understand what transformations like shifts and translations will do.
Asymptotes
Asymptotes are invisible lines that the graph of a rational function approaches but never reaches. They act as boundaries and guide the graph's behavior significantly. As for the transformation of the parent function \( y = \frac{1}{x} \), you will need to find new asymptotes for the function \( j(x) = \frac{3x - 7}{x - 2} \). Here's how we do it:
  • Vertical Asymptote: Set the denominator equal to zero and solve for \( x \):
    \( x - 2 = 0 \Rightarrow x = 2 \). This determines our vertical asymptote at \( x = 2 \).
  • Horizontal Asymptote: Evaluate the leading coefficients of the numerator and denominator:
    \( \frac{3}{1} = 3 \). This gives us the horizontal asymptote at \( y = 3 \).
These asymptotes reflect the lines the graph will approach as it extends infinitely in both directions. Knowing the asymptotes upfront is essential for sketching the graph correctly.
Horizontal Shift
A horizontal shift moves the graph of a function left or right on the coordinate plane. To identify this in the context of rational functions, focus on changes within the denominator. In our modified function \( j(x) = \frac{3x - 7}{x - 2} \), we observe a shift from \( x = 0 \) to \( x = 2 \). This occurs because:
  • The change \( x - 2 \) indicates a transformation 2 units to the right.
All points, including those on the hyperbola, are shifted to the right by 2 units as well. Recognizing this horizontal shift helps in plotting the new position of the function on a graph with respect to the parent function.
Vertical Translation
Vertical translation adjusts the graph either upward or downward on the coordinate plane. This occurs as a result of constants being added to or subtracted from the function. In your case, when analyzing \( j(x) = \frac{3x - 7}{x - 2} \), the presence of a horizontal asymptote at \( y=3 \) suggests a vertical translation:
  • The original horizontal asymptote \( y=0 \) shifts to \( y=3 \).
  • This indicates the entire graph is lifted upwards by 3 units.
This shift means that every point on the graph, along with the center of the hyperbola, is elevated by 3 units on the y-axis. Understanding vertical translations fine-tunes our comprehension of how the graph behaves relative to its initial form.

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