Quadratic Vertex Form

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Quadratic Vertex Form · Grade 10 · Worksheet 2

1. Charlotte is an engineer designing a suspension bridge. The main cable of the bridge hangs in the shape of a parabola. The height of the cable above the bridge deck, in meters, is modeled by the function h(x) = 0.02(x - 40)² + 10, where x is the horizontal distance from the left tower in meters. The vertex of this parabola represents the lowest point of the cable. What is the horizontal distance from the left tower to the lowest point of the cable, and what is the height of the cable at that point? Answer: Horizontal distance: 40 m, Height: 10 m Solution: The function is given in vertex form: h(x) = 0.02(x - 40)² + 10. In vertex form y = a(x - h)² + k, the vertex is (h, k). Here, h = 40 and k = 10.
   Full step-by-step solution

   Step 1: The function is given in vertex form: h(x) = 0.02(x - 40)² + 10. Step 2: In vertex form y = a(x - h)² + k, the vertex is (h, k). Here, h = 40 and k = 10. Step 3: The vertex is (40, 10). Since a = 0.02 is positive, the parabola opens upward, so the vertex represents the minimum point. Step 4: The x-coordinate of the vertex (40) is the horizontal distance from the left tower to the lowest point of the cable. Step 5: The y-coordinate of the vertex (10) is the height of the cable at that lowest point. Step 6: Therefore, the lowest point of the cable is 40 meters from the left tower, and its height is 10 meters above the bridge deck. The answer is: Horizontal distance: 40 m, Height: 10 m.

2. Convert f(x) = 6x² - 36x + 51 to vertex form f(x) = a(x - h)² + k = ? Answer: 6(x - 3)² - 3 Solution: Factor out 6 from the first two terms: f(x) = 6(x² - 6x) + 51 Complete the square inside the parentheses: Take half of -6, which is -3, square it to get 9.
   Full step-by-step solution

   Step 1: Factor out 6 from the first two terms: f(x) = 6(x² - 6x) + 51 Step 2: Complete the square inside the parentheses: Take half of -6, which is -3, square it to get 9. Step 3: Add and subtract 9 inside: f(x) = 6(x² - 6x + 9 - 9) + 51 Step 4: Rewrite as a perfect square: f(x) = 6[(x - 3)² - 9] + 51 Step 5: Distribute the 6: f(x) = 6(x - 3)² - 54 + 51 Step 6: Combine constants: f(x) = 6(x - 3)² - 3 The vertex form is f(x) = 6(x - 3)² - 3.

3. Convert f(x) = 4x² - 24x + 38 to vertex form f(x) = a(x - h)² + k = ? Answer: 4(x - 3)² + 2 Solution: Factor 4 from the first two terms: f(x) = 4(x² - 6x) + 38 Complete the square inside the parentheses: Take half of -6, which is -3, square it to get 9.
   Full step-by-step solution

   Step 1: Factor 4 from the first two terms: f(x) = 4(x² - 6x) + 38 Step 2: Complete the square inside the parentheses: Take half of -6, which is -3, square it to get 9. Step 3: Add and subtract 9 inside the parentheses: f(x) = 4(x² - 6x + 9 - 9) + 38 Step 4: Rewrite as a perfect square: f(x) = 4[(x - 3)² - 9] + 38 Step 5: Distribute the 4: f(x) = 4(x - 3)² - 36 + 38 Step 6: Combine constants: f(x) = 4(x - 3)² + 2 The vertex form is f(x) = 4(x - 3)² + 2.

4. y = 9(x - 12)^2 + 15. Find vertex and axis of symmetry. Answer: Vertex: (12, 15), Axis of symmetry: x = 12 Solution: Identify h and k from the equation y = 9(x - 12)^2 + 15. Here h = 12 and k = 15. The vertex is (h, k) = (12, 15).
   Full step-by-step solution

   Step 1: Identify h and k from the equation y = 9(x - 12)^2 + 15. Here h = 12 and k = 15. Step 2: The vertex is (h, k) = (12, 15). Step 3: The axis of symmetry is the vertical line x = h, so x = 12. The answer is Vertex: (12, 15), Axis of symmetry: x = 12.

5. Convert f(x) = -2x² + 16x - 35 to vertex form f(x) = a(x - h)² + k = ? Answer: -2(x - 4)² - 3 Solution: Factor -2 from the first two terms: f(x) = -2(x² - 8x) - 35 Complete the square inside the parentheses: Take half of -8, which is -4, square it to get 16 Add and subtract 16 inside the parentheses: f(x) = -2(x² - 8x + 16 - 16) - 35 Rewrite as: f(x) = -2[(x² - 8x + 16) - 16] - 35 Simplify: f(x) =…
   Full step-by-step solution

   Step 1: Factor -2 from the first two terms: f(x) = -2(x² - 8x) - 35 Step 2: Complete the square inside the parentheses: Take half of -8, which is -4, square it to get 16 Step 3: Add and subtract 16 inside the parentheses: f(x) = -2(x² - 8x + 16 - 16) - 35 Step 4: Rewrite as: f(x) = -2[(x² - 8x + 16) - 16] - 35 Step 5: Simplify: f(x) = -2[(x - 4)² - 16] - 35 Step 6: Distribute the -2: f(x) = -2(x - 4)² + 32 - 35 Step 7: Combine constants: f(x) = -2(x - 4)² - 3 The vertex form is f(x) = -2(x - 4)² - 3

6. Kaia is analyzing the graph of a parabola on a coordinate grid. The vertex of the parabola is at (7, 3). The parabola passes through the point (5, 11). Write the equation of the parabola in vertex form: y = a(x - h)^2 + k. Answer: y = 2(x - 7)^2 + 3 Solution: Identify h and k from the vertex (7, 3). So h = 7 and k = 3. Substitute into vertex form: y = a(x - 7)^2 + 3.
   Full step-by-step solution

   Step 1: Identify h and k from the vertex (7, 3). So h = 7 and k = 3. Step 2: Substitute into vertex form: y = a(x - 7)^2 + 3. Step 3: Use the point (5, 11) which lies on the parabola. Substitute x = 5 and y = 11: 11 = a(5 - 7)^2 + 3 Step 4: Simplify inside the parentheses: 5 - 7 = -2, so (-2)^2 = 4. 11 = a(4) + 3 Step 5: Subtract 3 from both sides: 8 = 4a Step 6: Divide by 4: a = 2 Step 7: Write the final equation: y = 2(x - 7)^2 + 3 The answer is y = 2(x - 7)^2 + 3.

7. Mason is designing a parabolic arch for a new garden entrance. The shape of the arch is modeled by the equation y = -2(x - 7)² + 22, where y is the height in feet and x is the horizontal distance from the left base of the arch. What is the vertex of the arch, and what is the axis of symmetry? Answer: Vertex: (7, 22), Axis of symmetry: x = 7 Solution: Identify the vertex form: y = a(x - h)² + k. The given equation is y = -2(x - 7)² + 22. Compare with vertex form: a = -2, h = 7, k = 22.
   Full step-by-step solution

   Step 1: Identify the vertex form: y = a(x - h)² + k. The given equation is y = -2(x - 7)² + 22. Step 2: Compare with vertex form: a = -2, h = 7, k = 22. Step 3: The vertex is at (h, k) = (7, 22). Step 4: The axis of symmetry is the vertical line through the vertex, x = h = 7. Final answer: Vertex: (7, 22), Axis of symmetry: x = 7.

8. A quadratic function in vertex form is given by f(x) = -2(x - 3)^2 + 8. What is the maximum value of this function? Answer: 8 Solution: We are given the quadratic function in vertex form: f(x) = -2(x - 3)^2 + 8 f(x) = a(x - h)^2 + k where (h, k) is the vertex of the parabola. From f(x) = -2(x - 3)^2 + 8: a = -2 h = 3 k = 8 So the vertex is (3, 8).
   Full step-by-step solution

   Let's solve step-by-step. We are given the quadratic function in vertex form: f(x) = -2(x - 3)^2 + 8 --- **Step 1: Understand vertex form** The vertex form of a quadratic is: f(x) = a(x - h)^2 + k where (h, k) is the vertex of the parabola. --- **Step 2: Identify a, h, k** From f(x) = -2(x - 3)^2 + 8: a = -2 h = 3 k = 8 So the vertex is (3, 8). --- **Step 3: Determine if the vertex is maximum or minimum** Since a = -2 is negative, the parabola opens downward. For a downward-opening parabola, the vertex is the maximum point. --- **Step 4: Find the maximum value** The maximum value of the function is the y-coordinate of the vertex, which is k = 8. --- **Step 5: Verify reasoning** The term -2(x - 3)^2 is always ≤ 0 because (x - 3)^2 ≥ 0 and we multiply by -2. So the largest value occurs when -2(x - 3)^2 = 0, which happens when x = 3. Then f(3) = 0 + 8 = 8. --- **Final Answer:** 8