Question

\- 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

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

Step-by-step solution

Function Rewrite

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

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.

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.

Rewrite and Simplify Correctly

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

Separate the Fraction

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

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.

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: \( y = -3 + \frac{23}{8-7} = 20\).

Verify with Technology

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

Key concepts

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.