Rational Roundup Worksheet
Rational Roundup
Algebra 2 Review: Rational Expressions & Functions
Name:
Section 1: Transformations
1
Transformation Mapping
Analyze the function \(f(x) = -2 + \frac{1}{x - 4}\). Describe the shifts from the parent function \(f(x) = \frac{1}{x}\), determine the equations for the new asymptotes, and sketch them clearly on the grid provided.
Horizontal Shift & Asymptote:
Vertical Shift & Asymptote:
Graphing Workspace (Sketch Asymptotes & Curves):
2 4 6 8 2 4 6 -2 -4 -6
Section 1 Continued...
2
Simplify the Product
Multiply and simplify the rational expressions below. Show all necessary factoring steps.
\[ \frac{5x^2 - 30x}{2x + 16} \cdot \frac{x + 8}{x - 6} \]
Simplified Result:
Section 2: Rational Modeling
3
Winning Streak
A success rate is modeled by \(y(x) = \frac{3 + x}{10 + x}\). Select all true statements below.
A. The vertical asymptote for this model occurs at \(x = -10\).
B. The horizontal asymptote for the success rate is at \(y = 1\).
C. Theoretically, the success rate will eventually reach 100%.
D. The horizontal asymptote represents the limit the rate approaches.
4
Field Trip Funding
Cost: $45 setup + $10 per shirt. Which function models average cost \(P(t)\)?
A. \(P(t) = \frac{10 + 45}{t}\)
B. \(P(t) = 10t + 45\)
C. \(P(t) = \frac{10t + 45}{t}\)
D. \(P(t) = \frac{10 + 45t}{t}\)
Work Space:
5
Combine the Fractions
Find the sum. Factor denominators completely to identify the LCD first.
\[ \frac{4x - 3}{x^2 + x - 2} + \frac{x}{x + 2} \]
Final Sum Result:
Section 3: Advanced Analysis
10
Asymptote Detective
Analyze the function below. Identify vertical asymptotes and holes, justifying your answers with work.
\[ f(x) = \frac{(x - 8)(2x + 3)(x + 4)(4x - 1)}{(x - 8)(x + 5)(2x - 1)} \]
Vertical Asymptotes & Work:
Removable Discontinuities (Holes) & Work:
District Standard Review Item 10
Rational Roundup Answer Key
Answer Key
Rational Roundup: Algebra 2 Review
Teacher Guide
1
Transformation Mapping
Horizontal:
Right 4 units \( (x = 4) \)
Vertical:
Down 2 units \( (y = -2) \)
Student graph should include dashed asymptotes at these locations.
2
Simplify the Product
\[ \frac{5x(x - 6)}{2(x + 8)} \cdot \frac{x + 8}{x - 6} = \frac{5x}{2} \]
Final Result: \( 2.5x \) or \( \frac{5x}{2} \)
3
Winning Streak
Correct Options: A, B, and D
Vertical Asymptote at \( x = -10 \), Horizontal Asymptote at \( y = 1 \). The rate approaches but mathematically never touches 100%.
4
Field Trip Funding
Correct: C
Models average cost: \( \frac{\text{Total Cost}}{\text{Quantity}} = \frac{10t + 45}{t} \)
5
Combine the Fractions
\[ \frac{4x - 3}{(x+2)(x-1)} + \frac{x}{x + 2} \]
Step 1: Identify the LCD
The first denominator factors to \( (x+2)(x-1) \). The second denominator is \( (x+2) \). Therefore, the Least Common Denominator is \( \mathbf{(x+2)(x-1)} \).
Step 2: Adjust & Combine
Multiply the second fraction by \( \frac{x-1}{x-1} \) to get common denominators, then combine the numerators:
\( (4x - 3) + x(x - 1) = 4x - 3 + x^2 - x \)
Final Sum: \( \frac{x^2 + 3x - 3}{(x+2)(x-1)} \)
Section 3: Advanced Analysis (District Standards alignment)
10
Asymptote Detective
Removable (Hole):
\( x = 8 \)
Why? The factor \( (x - 8) \) appears in both the numerator and denominator. Since it cancels out, the function is undefined at \( x = 8 \), but it creates a single missing point (hole) rather than an asymptote.
Vertical Asymptotes:
\( x = -5, x = 0.5 \)
Why? After canceling the common factor, we set the remaining denominator factors to zero:
\( x + 5 = 0 \Rightarrow x = -5 \)
\( 2x - 1 = 0 \Rightarrow x = 0.5 \)
Key Insight: Remind students that simplifying the expression *first* is the most critical step. If a root of the denominator also makes the numerator zero (like \( x = 8 \)), it indicates a removable discontinuity (hole). If it *only* makes the denominator zero (like \( x = -5 \)), it is a vertical asymptote.
