9 - 1 : Students will be able to determine the key features of reciprocal functions including: zeros, y-intercepts, vertical and horizontal asymptotes, domain, and range. 9 - 2 : Students will be able to graph absolute value functions and describe their transformations including: reflections across the x-axis, reflections across the y-axis, vertical stretch and compression, horizontal stretch and compression, horizontal shifts (left and right), and vertical shifts (up and down)
What is a Reciprocal Function
Key Features
Examples
Transformations
Graph the Reciprocal Function 9-1: Determine all Key Features of the function. 9-2: Describe all Transfomrations
Outcome 9-1 & 9-2
Graph the Reciprocal Function 9-1: Determine all Key Features of the function. 9-2: Describe all Transfomrations
Outcome 9-1 & 9-2
Graph the Reciprocal Function 9-1: Determine all Key Features of the function. 9-2: Describe all Transfomrations
Outcome 9-1 & 9-2
Zeros: (2.6, 0)y-intercept: (0, -2.6) V. Asymptote: x = 3 H. Asymptote: y = -3 Domain: (-∞, 3)U(3, ∞) Range: (-∞, -3)U(-3, ∞) End Behavior: As x → -∞, f(x) → -3 As x → 3, f(x) → ∞ As x → 3, f(x) → -∞ As x → ∞, f(x) → -3
1. Reflection: Reflection across x-axis 2. Stretch / Compression: Vertical Stretch (2) Horiztonal Stretch (1/2) 3. Horizontal Shift: Horizontal Shift Right (4) 4. Vertical Shift: Vertical Shift Up (1)
1. Reflection: No Reflection 2. Stretch / Compression: Vertical Stretch (4) Horizontal Compression (2) 3. Horizontal Shift: Horizontal Shift Left (2) 4. Vertical Shift: Vertical Shift Down (6)
Zeros: (3.2, 0)y-intercept: (0, 5.6) V. Asymptote: x = 3 H. Asymptote: y = 5 Domain: (-∞, 3)U(3, ∞) Range: (-∞, 5)U(5, ∞) End Behavior: As x → -∞, f(x) → 5 As x → 3, f(x) → ∞ As x → 3, f(x) → -∞ As x → ∞, f(x) → 5
1. Reflection: Reflection across y-axis 2. Stretch / Compression: Vertical Compression (1/4) Horizontal Stretch (2) 3. Horizontal Shift: Horizontal Shift Right (2) 4. Vertical Shift: Vertical Shift Down (3)
Zeros: (1.9, 0)y-intercept: (0, -2.9) V. Asymptote: x = 2 H. Asymptote: y = -3 Domain: (-∞, 2)U(2, ∞) Range: (-∞, -3)U(-3, ∞) End Behavior: As x → ∞, f(x) → -3 As x → 2, f(x) → ∞ As x → 2, f(x) → -∞ As x → ∞, f(x) → -3
1. Reflection: Reflection across x-axis 2. Stretch / Compression: Vertical Stretch (2) 3. Horizontal Shift: Horizontal Shift Right (3) 4. Vertical Shift: Vertical Shift Up (5)
1. Reflection: Reflection across x-axis 2. Stretch / Compression: No Stretch / Compression 3. Horizontal Shift: Horizontal Shift Left (3) 4. Vertical Shift: Vertical Shift Down (3)
Zeros: (1.5, 0)y-intercept: N/A V. Asymptote: x = 0 H. Asymptote: y = 4 Domain: (-∞, 0)U(0, ∞) Range: (-∞, 4)U(4, ∞) End Behavior: As x → -∞, f(x) → 4 As x → 0, f(x) → ∞ As x → 0, f(x) → -∞ As x → ∞, f(x) → 4
Zeros: (-1.6, 0)y-intercept: (0, -5) V. Asymptote: x = -2 H. Asymptote: y = -6 Domain: (-∞, -2)U(-2, ∞) Range: (-∞, -6)U(-6, ∞) End Behavior: As x → -∞, f(x) → -6 As x → -2, f(x) → -∞ As x → -2, f(x) → ∞ As x → ∞, f(x) → -6
Zeros: (8, 0)y-intercept: (0, 2) V. Asymptote: x = 4 H. Asymptote: y = 1 Domain: (-∞, 4)U(4, ∞) Range: (-∞, 1)U(1, ∞) End Behavior: As x → -∞, f(x) → 1 As x → 4, f(x) → ∞ As x → 4, f(x) → -∞ As x → ∞, f(x) → 1
1. Reflection: Reflection across y-axis 2. Stretch / Compression: Vertical Stretch (2) 3. Horizontal Shift: No Horizontal Shift 4. Vertical Shift: Vertical Shift Up (4)
Rational Functions: Graphing - Reciprocal Functions
Kevin Helms
Created on September 2, 2025
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Transcript
9 - 1 : Students will be able to determine the key features of reciprocal functions including: zeros, y-intercepts, vertical and horizontal asymptotes, domain, and range. 9 - 2 : Students will be able to graph absolute value functions and describe their transformations including: reflections across the x-axis, reflections across the y-axis, vertical stretch and compression, horizontal stretch and compression, horizontal shifts (left and right), and vertical shifts (up and down)
What is a Reciprocal Function
Key Features
Examples
Transformations
Graph the Reciprocal Function 9-1: Determine all Key Features of the function. 9-2: Describe all Transfomrations
Outcome 9-1 & 9-2
Graph the Reciprocal Function 9-1: Determine all Key Features of the function. 9-2: Describe all Transfomrations
Outcome 9-1 & 9-2
Graph the Reciprocal Function 9-1: Determine all Key Features of the function. 9-2: Describe all Transfomrations
Outcome 9-1 & 9-2
Zeros: (2.6, 0)y-intercept: (0, -2.6) V. Asymptote: x = 3 H. Asymptote: y = -3 Domain: (-∞, 3)U(3, ∞) Range: (-∞, -3)U(-3, ∞) End Behavior: As x → -∞, f(x) → -3 As x → 3, f(x) → ∞ As x → 3, f(x) → -∞ As x → ∞, f(x) → -3
1. Reflection: Reflection across x-axis 2. Stretch / Compression: Vertical Stretch (2) Horiztonal Stretch (1/2) 3. Horizontal Shift: Horizontal Shift Right (4) 4. Vertical Shift: Vertical Shift Up (1)
1. Reflection: No Reflection 2. Stretch / Compression: Vertical Stretch (4) Horizontal Compression (2) 3. Horizontal Shift: Horizontal Shift Left (2) 4. Vertical Shift: Vertical Shift Down (6)
Zeros: (3.2, 0)y-intercept: (0, 5.6) V. Asymptote: x = 3 H. Asymptote: y = 5 Domain: (-∞, 3)U(3, ∞) Range: (-∞, 5)U(5, ∞) End Behavior: As x → -∞, f(x) → 5 As x → 3, f(x) → ∞ As x → 3, f(x) → -∞ As x → ∞, f(x) → 5
1. Reflection: Reflection across y-axis 2. Stretch / Compression: Vertical Compression (1/4) Horizontal Stretch (2) 3. Horizontal Shift: Horizontal Shift Right (2) 4. Vertical Shift: Vertical Shift Down (3)
Zeros: (1.9, 0)y-intercept: (0, -2.9) V. Asymptote: x = 2 H. Asymptote: y = -3 Domain: (-∞, 2)U(2, ∞) Range: (-∞, -3)U(-3, ∞) End Behavior: As x → ∞, f(x) → -3 As x → 2, f(x) → ∞ As x → 2, f(x) → -∞ As x → ∞, f(x) → -3
1. Reflection: Reflection across x-axis 2. Stretch / Compression: Vertical Stretch (2) 3. Horizontal Shift: Horizontal Shift Right (3) 4. Vertical Shift: Vertical Shift Up (5)
1. Reflection: Reflection across x-axis 2. Stretch / Compression: No Stretch / Compression 3. Horizontal Shift: Horizontal Shift Left (3) 4. Vertical Shift: Vertical Shift Down (3)
Zeros: (1.5, 0)y-intercept: N/A V. Asymptote: x = 0 H. Asymptote: y = 4 Domain: (-∞, 0)U(0, ∞) Range: (-∞, 4)U(4, ∞) End Behavior: As x → -∞, f(x) → 4 As x → 0, f(x) → ∞ As x → 0, f(x) → -∞ As x → ∞, f(x) → 4
Zeros: (-1.6, 0)y-intercept: (0, -5) V. Asymptote: x = -2 H. Asymptote: y = -6 Domain: (-∞, -2)U(-2, ∞) Range: (-∞, -6)U(-6, ∞) End Behavior: As x → -∞, f(x) → -6 As x → -2, f(x) → -∞ As x → -2, f(x) → ∞ As x → ∞, f(x) → -6
Zeros: (8, 0)y-intercept: (0, 2) V. Asymptote: x = 4 H. Asymptote: y = 1 Domain: (-∞, 4)U(4, ∞) Range: (-∞, 1)U(1, ∞) End Behavior: As x → -∞, f(x) → 1 As x → 4, f(x) → ∞ As x → 4, f(x) → -∞ As x → ∞, f(x) → 1
1. Reflection: Reflection across y-axis 2. Stretch / Compression: Vertical Stretch (2) 3. Horizontal Shift: No Horizontal Shift 4. Vertical Shift: Vertical Shift Up (4)