Quadratic Functions

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Quadratic Functions · Grade 9 · Worksheet 3

1. Isabella is designing a custom frame for a painting. The frame's area can be modeled by the quadratic function A(w) = 3(w - 9)² + 48, where w is the width in inches. Convert this function to standard form A(w) = aw² + bw + c to determine the coefficient of the linear term. Answer: -54 Solution: Start with the vertex form: A(w) = 3(w - 9)² + 48 Expand the squared binomial: (w - 9)² = w² - 18w + 81 Multiply by the coefficient 3: 3(w² - 18w + 81) = 3w² - 54w + 243 Add the constant term: 3w² - 54w + 243 + 48 = 3w² - 54w + 291 The standard form is A(w) = 3w² - 54w + 291, so the coefficient…
   Full step-by-step solution

   Step 1: Start with the vertex form: A(w) = 3(w - 9)² + 48 Step 2: Expand the squared binomial: (w - 9)² = w² - 18w + 81 Step 3: Multiply by the coefficient 3: 3(w² - 18w + 81) = 3w² - 54w + 243 Step 4: Add the constant term: 3w² - 54w + 243 + 48 = 3w² - 54w + 291 Step 5: The standard form is A(w) = 3w² - 54w + 291, so the coefficient of the linear term is -54.

2. 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

3. Convert y = 2(x - 4)² + 6 to standard form Answer: y = 2x² - 16x + 38 Solution: Start with y = 2(x - 4)² + 6 Expand (x - 4)² = x² - 8x + 16 Multiply by 2: 2(x² - 8x + 16) = 2x² - 16x + 32 Add the constant term: 2x² - 16x + 32 + 6 Combine like terms: 2x² - 16x + 38 The answer is y = 2x² - 16x + 38.
   Full step-by-step solution

   Step 1: Start with y = 2(x - 4)² + 6 Step 2: Expand (x - 4)² = x² - 8x + 16 Step 3: Multiply by 2: 2(x² - 8x + 16) = 2x² - 16x + 32 Step 4: Add the constant term: 2x² - 16x + 32 + 6 Step 5: Combine like terms: 2x² - 16x + 38 The answer is y = 2x² - 16x + 38.

4. Convert y = 5(x - 3)² + 7 to standard form Answer: y = 5x² - 30x + 52 Solution: Start with y = 5(x - 3)² + 7 Expand (x - 3)² = x² - 6x + 9 Multiply by 5: 5(x² - 6x + 9) = 5x² - 30x + 45 Add the constant term: 5x² - 30x + 45 + 7 Combine like terms: 5x² - 30x + 52 The answer is y = 5x² - 30x + 52
   Full step-by-step solution

   Step 1: Start with y = 5(x - 3)² + 7 Step 2: Expand (x - 3)² = x² - 6x + 9 Step 3: Multiply by 5: 5(x² - 6x + 9) = 5x² - 30x + 45 Step 4: Add the constant term: 5x² - 30x + 45 + 7 Step 5: Combine like terms: 5x² - 30x + 52 The answer is y = 5x² - 30x + 52

5. Aroha is analyzing the path of a soccer ball kicked during practice. The ball's height above the ground follows the equation y = 3(x - 5)² + 7, where y is the height in meters and x is the horizontal distance in meters from where Aroha kicked it. What is the height of the ball when it is directly above the point where Aroha kicked it? Answer: 82 Solution: The ball is directly above the point where Aroha kicked it when the horizontal distance x = 0.
   Full step-by-step solution

   Step 1: The ball is directly above the point where Aroha kicked it when the horizontal distance x = 0. Step 2: Substitute x = 0 into the equation y = 3(x - 5)² + 7 Step 3: y = 3(0 - 5)² + 7 Step 4: y = 3(-5)² + 7 Step 5: y = 3(25) + 7 Step 6: y = 75 + 7 Step 7: y = 82 Step 8: The height of the ball is 82 meters.

6. Emma is analyzing the profit from her lemonade stand using a quadratic model. Her profit function in standard form is P(x) = -3x² + 24x - 13, where x represents the price per cup in dollars. What is the maximum profit Emma can achieve? Answer: 35 Solution: Identify the coefficients from P(x) = -3x² + 24x - 13 a = -3, b = 24, c = -13 Find the x-coordinate of the vertex using x = -b/(2a) x = -24/(2×(-3)) = -24/(-6) = 4 Substitute x = 4 back into the function to find the maximum profit P(4) = -3(4)² + 24(4) - 13 P(4) = -3(16) + 96 - 13 P(4) = -48 +…
   Full step-by-step solution

   Step 1: Identify the coefficients from P(x) = -3x² + 24x - 13 a = -3, b = 24, c = -13 Step 2: Find the x-coordinate of the vertex using x = -b/(2a) x = -24/(2×(-3)) = -24/(-6) = 4 Step 3: Substitute x = 4 back into the function to find the maximum profit P(4) = -3(4)² + 24(4) - 13 P(4) = -3(16) + 96 - 13 P(4) = -48 + 96 - 13 P(4) = 48 - 13 P(4) = 35 The maximum profit Emma can achieve is $35.

7. Aroha is designing a decorative fountain where the water follows a parabolic path. The height of the water stream in meters is given by the equation y = -2(x - 5)² + 18. What is the maximum height the water reaches? Answer: 18 Solution: The equation is given in vertex form: y = a(x - h)² + k For Aroha's fountain, the equation is y = -2(x - 5)² + 18 In vertex form, the vertex is at (h, k) = (5, 18) Since the coefficient a = -2 is negative, the parabola opens downward When a parabola opens downward, the vertex represents the…
   Full step-by-step solution

   Step 1: The equation is given in vertex form: y = a(x - h)² + k Step 2: For Aroha's fountain, the equation is y = -2(x - 5)² + 18 Step 3: In vertex form, the vertex is at (h, k) = (5, 18) Step 4: Since the coefficient a = -2 is negative, the parabola opens downward Step 5: When a parabola opens downward, the vertex represents the maximum point Step 6: Therefore, the maximum height is the y-coordinate of the vertex, which is 18 The maximum height the water reaches is 18 meters.

8. Convert y = -2(x - 5)² + 10 to standard form Answer: y = -2x² + 20x - 40 Solution: Start with y = -2(x - 5)² + 10 Expand (x - 5)² = x² - 10x + 25 Multiply by -2: -2(x² - 10x + 25) = -2x² + 20x - 50 Add the constant +10: -2x² + 20x - 50 + 10 Combine like terms: -2x² + 20x - 40 The standard form is y = -2x² + 20x - 40
   Full step-by-step solution

   Step 1: Start with y = -2(x - 5)² + 10 Step 2: Expand (x - 5)² = x² - 10x + 25 Step 3: Multiply by -2: -2(x² - 10x + 25) = -2x² + 20x - 50 Step 4: Add the constant +10: -2x² + 20x - 50 + 10 Step 5: Combine like terms: -2x² + 20x - 40 The standard form is y = -2x² + 20x - 40

9. Convert y = -3(x - 7)² + 11 to standard form y = ax² + bx + c Answer: y = -3x² + 42x - 136 Solution: Expand (x - 7)² = x² - 14x + 49 Multiply by -3: -3(x² - 14x + 49) = -3x² + 42x - 147 Add the constant term: -3x² + 42x - 147 + 11 Combine like terms: -3x² + 42x - 136 The standard form is y = -3x² + 42x - 136
   Full step-by-step solution

   Step 1: Expand (x - 7)² = x² - 14x + 49 Step 2: Multiply by -3: -3(x² - 14x + 49) = -3x² + 42x - 147 Step 3: Add the constant term: -3x² + 42x - 147 + 11 Step 4: Combine like terms: -3x² + 42x - 136 The standard form is y = -3x² + 42x - 136