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Quadratic Functions

Grade 9 · Algebra · Worksheet 2

  1. Convert y = 4(x - 7)² + 9 to standard form y = ax² + bx + c Answer: ______________
  2. Mere is analyzing a quadratic function in vertex form: y = 2(x - 4)² + 6. She wants to convert it to standard form y = ax² + bx + c. What is the value of the coefficient b in the standard form? Answer: ______________
  3. Convert y = 2(x - 7)² + 9 to standard form Answer: ______________
  4. Convert y = -2(x - 7)² + 11 to standard form Answer: ______________
  5. Convert y = 2(x - 7)² + 8 to standard form Answer: ______________
  6. Convert y = -3(x + 5)² + 10 to standard form Answer: ______________
  7. Convert y = -3(x - 1)² + 6 to standard form Answer: ______________
  8. Isabella is designing a custom picture frame for her art project. The frame's border follows the quadratic function y = 2(x - 8)² + 10, where y represents the border width in centimeters and x represents the distance from the left edge of the frame. She needs to convert this equation to standard form to program her cutting machine. What is the coefficient of the x term when the equation is written in standard form y = ax² + bx + c? Answer: ______________
  9. Kaia is designing a custom frame for a painting. The frame's area can be modeled by the quadratic function A(w) = -2w² + 28w - 80, where w is the width in centimeters. What is the maximum possible area of the frame in square centimeters? Answer: ______________
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Answer Key & Explanations

Quadratic Functions · Grade 9 · Worksheet 2

  1. Convert y = 4(x - 7)² + 9 to standard form y = ax² + bx + c Answer: y = 4x² - 56x + 205 Solution: Start with y = 4(x - 7)² + 9 Expand (x - 7)² using the formula (a - b)² = a² - 2ab + b² (x - 7)² = x² - 2(x)(7) + 7² = x² - 14x + 49 Multiply by 4: 4(x² - 14x + 49) = 4x² - 56x + 196 Add the constant term: 4x² - 56x + 196 + 9 = 4x² - 56x + 205 Write in standard form: y = 4x² - 56x + 205
    Full step-by-step solution

    Step 1: Start with y = 4(x - 7)² + 9 Step 2: Expand (x - 7)² using the formula (a - b)² = a² - 2ab + b² (x - 7)² = x² - 2(x)(7) + 7² = x² - 14x + 49 Step 3: Multiply by 4: 4(x² - 14x + 49) = 4x² - 56x + 196 Step 4: Add the constant term: 4x² - 56x + 196 + 9 = 4x² - 56x + 205 Step 5: Write in standard form: y = 4x² - 56x + 205

  2. Mere is analyzing a quadratic function in vertex form: y = 2(x - 4)² + 6. She wants to convert it to standard form y = ax² + bx + c. What is the value of the coefficient b in the standard form? Answer: -16 Solution: Start with the vertex form: y = 2(x - 4)² + 6 Expand the squared term: (x - 4)² = x² - 8x + 16 Multiply by the coefficient 2: 2(x² - 8x + 16) = 2x² - 16x + 32 Add the constant term: y = 2x² - 16x + 32 + 6 Combine like terms: y = 2x² - 16x + 38 The coefficient b is the number multiplied by x,…
    Full step-by-step solution

    Step 1: Start with the vertex form: y = 2(x - 4)² + 6 Step 2: Expand the squared term: (x - 4)² = x² - 8x + 16 Step 3: Multiply by the coefficient 2: 2(x² - 8x + 16) = 2x² - 16x + 32 Step 4: Add the constant term: y = 2x² - 16x + 32 + 6 Step 5: Combine like terms: y = 2x² - 16x + 38 Step 6: The coefficient b is the number multiplied by x, which is -16 Therefore, the value of b is -16.

  3. Convert y = 2(x - 7)² + 9 to standard form Answer: y = 2x² - 28x + 107 Solution: Start with y = 2(x - 7)² + 9 Expand (x - 7)² = x² - 14x + 49 Multiply by 2: 2(x² - 14x + 49) = 2x² - 28x + 98 Add the constant term: 2x² - 28x + 98 + 9 Combine like terms: 2x² - 28x + 107 The answer is y = 2x² - 28x + 107.
    Full step-by-step solution

    Step 1: Start with y = 2(x - 7)² + 9 Step 2: Expand (x - 7)² = x² - 14x + 49 Step 3: Multiply by 2: 2(x² - 14x + 49) = 2x² - 28x + 98 Step 4: Add the constant term: 2x² - 28x + 98 + 9 Step 5: Combine like terms: 2x² - 28x + 107 The answer is y = 2x² - 28x + 107.

  4. Convert y = -2(x - 7)² + 11 to standard form Answer: y = -2x² + 28x - 87 Solution: Start with y = -2(x - 7)² + 11 Expand (x - 7)² = x² - 14x + 49 Multiply by -2: -2(x² - 14x + 49) = -2x² + 28x - 98 Add the constant term: -2x² + 28x - 98 + 11 Combine like terms: -2x² + 28x - 87 The final answer is y = -2x² + 28x - 87
    Full step-by-step solution

    Step 1: Start with y = -2(x - 7)² + 11 Step 2: Expand (x - 7)² = x² - 14x + 49 Step 3: Multiply by -2: -2(x² - 14x + 49) = -2x² + 28x - 98 Step 4: Add the constant term: -2x² + 28x - 98 + 11 Step 5: Combine like terms: -2x² + 28x - 87 The final answer is y = -2x² + 28x - 87

  5. Convert y = 2(x - 7)² + 8 to standard form Answer: y = 2x² - 28x + 106 Solution: Start with the vertex form: y = 2(x - 7)² + 8 Expand (x - 7)² using (a - b)² = a² - 2ab + b² (x - 7)² = x² - 2(x)(7) + 7² = x² - 14x + 49 Multiply by the coefficient 2: 2(x² - 14x + 49) = 2x² - 28x + 98 Add the constant term: 2x² - 28x + 98 + 8 = 2x² - 28x + 106 The standard form is y = 2x² -…
    Full step-by-step solution

    Step 1: Start with the vertex form: y = 2(x - 7)² + 8 Step 2: Expand (x - 7)² using (a - b)² = a² - 2ab + b² (x - 7)² = x² - 2(x)(7) + 7² = x² - 14x + 49 Step 3: Multiply by the coefficient 2: 2(x² - 14x + 49) = 2x² - 28x + 98 Step 4: Add the constant term: 2x² - 28x + 98 + 8 = 2x² - 28x + 106 Step 5: The standard form is y = 2x² - 28x + 106

  6. Convert y = -3(x + 5)² + 10 to standard form Answer: y = -3x² - 30x - 65 Solution: Start with y = -3(x + 5)² + 10 Expand (x + 5)² = x² + 10x + 25 Multiply by -3: -3(x² + 10x + 25) = -3x² - 30x - 75 Add the constant term: -3x² - 30x - 75 + 10 Combine like terms: -3x² - 30x - 65 The answer is y = -3x² - 30x - 65
    Full step-by-step solution

    Step 1: Start with y = -3(x + 5)² + 10 Step 2: Expand (x + 5)² = x² + 10x + 25 Step 3: Multiply by -3: -3(x² + 10x + 25) = -3x² - 30x - 75 Step 4: Add the constant term: -3x² - 30x - 75 + 10 Step 5: Combine like terms: -3x² - 30x - 65 The answer is y = -3x² - 30x - 65

  7. Convert y = -3(x - 1)² + 6 to standard form Answer: y = -3x² + 6x + 3 Solution: Expand (x - 1)² = x² - 2x + 1 Multiply by -3: -3(x² - 2x + 1) = -3x² + 6x - 3 Add the constant +6: -3x² + 6x - 3 + 6 Combine like terms: -3x² + 6x + 3 The standard form is y = -3x² + 6x + 3
    Full step-by-step solution

    Step 1: Expand (x - 1)² = x² - 2x + 1 Step 2: Multiply by -3: -3(x² - 2x + 1) = -3x² + 6x - 3 Step 3: Add the constant +6: -3x² + 6x - 3 + 6 Step 4: Combine like terms: -3x² + 6x + 3 The standard form is y = -3x² + 6x + 3

  8. Isabella is designing a custom picture frame for her art project. The frame's border follows the quadratic function y = 2(x - 8)² + 10, where y represents the border width in centimeters and x represents the distance from the left edge of the frame. She needs to convert this equation to standard form to program her cutting machine. What is the coefficient of the x term when the equation is written in standard form y = ax² + bx + c? Answer: -32 Solution: Start with the vertex form: y = 2(x - 8)² + 10 Expand the squared binomial: (x - 8)² = x² - 16x + 64 Distribute the coefficient 2: 2(x² - 16x + 64) = 2x² - 32x + 128 Add the constant term: 2x² - 32x + 128 + 10 = 2x² - 32x + 138 The equation in standard form is y = 2x² - 32x + 138 The coefficient…
    Full step-by-step solution

    Step 1: Start with the vertex form: y = 2(x - 8)² + 10 Step 2: Expand the squared binomial: (x - 8)² = x² - 16x + 64 Step 3: Distribute the coefficient 2: 2(x² - 16x + 64) = 2x² - 32x + 128 Step 4: Add the constant term: 2x² - 32x + 128 + 10 = 2x² - 32x + 138 Step 5: The equation in standard form is y = 2x² - 32x + 138 Step 6: The coefficient of the x term is -32 The answer is -32.

  9. Kaia is designing a custom frame for a painting. The frame's area can be modeled by the quadratic function A(w) = -2w² + 28w - 80, where w is the width in centimeters. What is the maximum possible area of the frame in square centimeters? Answer: 18 Solution: Identify the coefficients: a = -2, b = 28, c = -80 Find the x-coordinate of the vertex using h = -b/(2a) h = -28/(2×-2) = -28/-4 = 7 Substitute w = 7 into the function to find the maximum area A(7) = -2(7)² + 28(7) - 80 A(7) = -2(49) + 196 - 80 A(7) = -98 + 196 - 80 A(7) = 98 - 80 A(7) = 18 The…
    Full step-by-step solution

    Step 1: Identify the coefficients: a = -2, b = 28, c = -80 Step 2: Find the x-coordinate of the vertex using h = -b/(2a) h = -28/(2×-2) = -28/-4 = 7 Step 3: Substitute w = 7 into the function to find the maximum area A(7) = -2(7)² + 28(7) - 80 A(7) = -2(49) + 196 - 80 A(7) = -98 + 196 - 80 A(7) = 98 - 80 A(7) = 18 Step 4: The maximum possible area is 18 square centimeters.