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Chapter 9: Problem 10

\- Mira uses algebra to rewrite the function \(y=\frac{2-3 x}{x-7}\) in an equivalent form that she can graph by hand. \(y=\frac{2-3 x}{x-7}\) \(y=\frac{-3 x+2}{x-7}\) \(y=\frac{-3 x-21+21+2}{x-7}\) \(y=\frac{-3(x-7)+23}{x-7}\) \(y=\frac{-3(x-7)}{x-7}+\frac{23}{x-7}\) \(y=-3+\frac{23}{x-7}\) \(y=\frac{23}{x-7}-3\) a) Identify and correct any errors in Mira's work. b) How might Mira have discovered that she had made an error without using technology? How might she have done so with technology?

Short Answer

Expert verified
Mira made an unneeded rearrangement in step 3. Correctly simplified remains \( y = -3 + \frac{23}{x-7} \).

Step by step solution

01

Function Rewrite

Starting from the original function: \( y = \frac{2 - 3x}{x - 7} \)
02

Check Algebraic Operations

Verify Mira's rewrittens step: \( y = \frac{-3x + 2}{x - 7} \) Notice that this rewriting is simply a rearrangement and is correct.
03

Break Down the Numerator

Expand and break down the numerator: \( y = \frac{-3x - 21 + 21 + 2}{x - 7} \)Combine \-21 + 21\ (they cancel each other out) and this outer part fails logical simplification.
04

Rewrite and Simplify Correctly

Rewrite without unnecessary terms: \( y = \frac{-3(x - 7) + 23}{x-7} \) is correct for breaking down \(2\) as \(-21 + 23\).
05

Separate the Fraction

Next, separate the fraction: \( y = \frac{-3(x - 7)}{x - 7} + \frac{23}{x - 7} \)Simplify the first fraction: \( y = -3 + \frac{23}{x - 7} \).
06

Graph the Function Manually

To graph manually, recognize \( y = -3 + \frac{23}{x- 7}\) as a transformed rational function. Shift starting graph of \( \frac{1}{x} \) horizontally by 7 and vertically by -3.
07

Check for Errors Manually

Check for errors by substituting values; for example, let \(x = 8\). Original: \( y = \frac{2 - 3(8)}{8 - 7} = -22 \) whereas simplified: \( y = -3 + \frac{23}{8-7} = 20\).
08

Verify with Technology

Use a graphing tool or software. Both equations should match the plotted curves completely.

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Key Concepts

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

Function Simplification
Simplifying functions can make graphing easier and help identify key features, such as asymptotes and intercepts. Let's understand Mira's approach to simplifying the function.First, she starts with the function: \( y = \frac{2 - 3x}{x - 7} \)She rearranges it to: \( y = \frac{-3x + 2}{x - 7} \)Though the order of terms in the numerator changes, the function itself remains the same, ensuring correctness. However, her next steps involve questionable algebraic manipulation which can lead to confusion: \( y = \frac{-3x - 21 + 21 + 2}{x - 7} \)Adding and subtracting -21 and 21 appears unnecessary here. Instead, we should break down the numerator clearly. The correct approach would be to rewrite the numerator in a way that matches the denominator without creating unnecessary steps.Thus: \( y = \frac{-3(x - 7) + 23}{x - 7} \)Separation of terms is correctly done in the next step: \( y = \frac{-3(x - 7)}{x - 7} + \frac{23}{x - 7} \)Which simplifies to: \( y = -3 + \frac{23}{x - 7} \)Now, the simplified form \( y = -3 + \frac{23}{x - 7} \) reveals a horizontal shift right by 7 units and a vertical shift down by 3 units, making the function easier to graph.
Graphing Rational Functions
Graphing rational functions involves understanding their structure and transformations. For the function \( y = -3 + \frac{23}{x - 7} \), let's identify graphing steps.
  • Recognize the basic form: This is based on the hyperbola \( \frac{1}{x} \).
  • Horizontal Shift: The term \( x - 7 \) indicates a shift 7 units to the right.
  • Vertical Shift: The term \( -3 \) means shifting the entire graph 3 units downward.
  • Vertical Asymptote: The denominator \( x - 7 \) becomes zero at \( x = 7 \), causing a vertical asymptote.
  • Horizontal Asymptote: For large values of \( x \), \( \frac{23}{x - 7} \) approaches zero, making \( y = -3 \) the horizontal asymptote.
Graphing involves first plotting the asymptotes, then sketching the behavior around these lines. For rational functions, note the curve's approach to asymptotes without intersecting them. The function \( y = -3 + \frac{23}{x - 7} \) would thus reflect these transformations, providing a clear graph.
Error Checking in Algebra
Ensuring correctness in algebraic manipulation is crucial. Mira can identify and correct errors by double-checking her steps and using basic principles.
  • Manual Verification: Substitute a value for \( x \) and compare outcomes. For example, when \( x = 8 \): \( y = \frac{2 - 3 \times 8}{8 - 7} = -22 \) vs. \( y = -3 + \frac{23}{8 - 7} = 20 \). The mismatch reveals an error in simplification.
  • Cross-checking Simplifications: Regularly return to the original expression to ensure consistency post-simplification.
  • Using Technology: Tools like graphing calculators or software are invaluable. By graphing the original and simplified functions, consistent plots confirm correctness. Discrepancies signal errors in algebra.
Systematic re-evaluation of each transformation helps catch mistakes. Whether manually comparing values or using technology, continual practice hones error-checking skills.

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Most popular questions from this chapter

Ryan and Kandra are kayaking near Lowe Inlet Marine Provincial Park on Grenville Channel, British Columbia. The current can flow in either direction at up to \(4 \mathrm{km} / \mathrm{h}\) depending on tidal conditions. Ryan and Kandra are capable of kayaking steadily at \(4 \mathrm{km} / \mathrm{h}\) without the current. a) What function relates the time, \(t,\) in hours, it will take them to travel \(4 \mathrm{km}\) along the channel as a function of the speed, \(w\), in kilometres per hour, of the current? What domain is possible for \(w\) in this context? b) Graph the function for an appropriate domain. c) Explain the behaviour of the graph for values at and near its non- permissible value and what the behaviour means in this situation.

The function \(h(v)=\frac{6378 v^{2}}{125-v^{2}}\) gives the maximum height, \(h,\) in kilometres, as a function of the initial velocity, \(v,\) in kilometres per second, for an object launched upward from Earth's surface, if the object gets no additional propulsion and air resistance is ignored. a) Graph the function. What parts of the graph are applicable to this situation? b) Explain what the graph indicates about how the maximum height is affected by the initial velocity. c) The term escape velocity refers to the initial speed required to break free of a gravitational field. Describe the nature of the graph for its non- permissible value, and explain why it represents the escape velocity for the object.

a) Predict the shape of the graph of \(y=\frac{2 x^{2}+2}{x^{2}-1}\) and explain your reasoning. b) Use graphing technology to confirm your prediction. c) How would the graph of each of the following functions compare to the one in part a)? Check using graphing technology. i) \(y=\frac{2 x^{2}-2}{x^{2}-1}\) ii) \(y=\frac{2 x^{2}+2}{x^{2}+1}\)

Solve each equation algebraically. a) \(\frac{5 x}{3 x+4}=7\) b) \(2=\frac{20-3 x}{x}\) c) \(\frac{x^{2}}{x-2}=x-6\) d) \(1+\frac{2}{x}=\frac{x}{x+3}\)

Hanna is shopping for a new deep freezer and is deciding between two models. One model costs \(\$ 500\) and has an estimated electricity cost of S100/year. A second model that is more energy efficient costs S800 but has an estimated electricity cost of \(\$ 60 /\) year. a) For each freezer, write an equation for the average cost per year as a function of the time, in years. b) Graph the functions for a reasonable domain. c) Identify important characteristics of each graph and explain what they show about the situation. d) How can the graph help Hanna decide which model to choose?
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