Quadratic Vertex Form

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Answer Key & Explanations

Quadratic Vertex Form · Grade 10 · Worksheet 1

1. Mere is designing a parabolic arch for a new garden entrance. The arch's shape is modeled by the quadratic function y = -2(x - 4)² + 18, where y is the height in meters above the ground and x is the horizontal distance in meters from the left edge of the arch. The local council requires that the highest point of the arch be at least 16 meters high for clearance. Determine the vertex of the arch and state whether it meets the council's height requirement. Answer: Vertex is (4, 18), which meets the requirement because 18 > 16. Solution: The function is given in vertex form y = a(x - h)² + k, where (h, k) is the vertex. Here, a = -2, h = 4, k = 18. The vertex is at (4, 18).
   Full step-by-step solution

   Step 1: The function is given in vertex form y = a(x - h)² + k, where (h, k) is the vertex. Here, a = -2, h = 4, k = 18. Step 2: The vertex is at (4, 18). This means the highest point of the arch is 18 meters above the ground. Step 3: The council requires the highest point to be at least 16 meters. Since 18 is greater than 16, the arch meets the requirement. Final answer: Vertex is (4, 18), and the arch meets the council's height requirement.

2. Mere is designing a water fountain for a new park. The height of the water stream (in meters) above the fountain nozzle is modeled by the function h(d) = -0.04(d - 25)^2 + 30, where d is the horizontal distance (in meters) from the nozzle. The park committee wants to know the maximum height the water reaches and the horizontal distance from the nozzle where this occurs. What are these two values? Answer: Maximum height = 30 m at horizontal distance = 25 m Solution: Identify the vertex form: h(d) = -0.04(d - 25)^2 + 30. Here, a = -0.04, h = 25, k = 30. The vertex of a parabola in vertex form y = a(x - h)^2 + k is (h, k).
   Full step-by-step solution

   Step 1: Identify the vertex form: h(d) = -0.04(d - 25)^2 + 30. Here, a = -0.04, h = 25, k = 30. Step 2: The vertex of a parabola in vertex form y = a(x - h)^2 + k is (h, k). So the vertex is (25, 30). Step 3: Since a = -0.04 is negative, the parabola opens downward, so the vertex represents the maximum point. Step 4: The x-coordinate (d = 25) is the horizontal distance from the nozzle where the maximum height occurs. Step 5: The y-coordinate (k = 30) is the maximum height of the water stream. Final answer: The water reaches its maximum height of 30 meters at a horizontal distance of 25 meters from the nozzle.

3. A quadratic function in vertex form is given by f(x) = 3(x + 4)^2 - 27. What is the minimum value of this function? Answer: -27 Solution: The function is in vertex form f(x) = a(x - h)^2 + k, where (h,k) is the vertex Our function is f(x) = 3(x + 4)^2 - 27, which can be rewritten as f(x) = 3(x - (-4))^2 + (-27) This means the vertex is at (-4, -27) Since a = 3 is positive, the parabola opens upward, so the vertex represents the…
   Full step-by-step solution

   Step 1: The function is in vertex form f(x) = a(x - h)^2 + k, where (h,k) is the vertex Step 2: Our function is f(x) = 3(x + 4)^2 - 27, which can be rewritten as f(x) = 3(x - (-4))^2 + (-27) Step 3: This means the vertex is at (-4, -27) Step 4: Since a = 3 is positive, the parabola opens upward, so the vertex represents the minimum point Step 5: The y-coordinate of the vertex gives the minimum value of the function Step 6: Therefore, the minimum value is -27 The answer is -27.

4. Olivia is a park designer creating a decorative fountain. The water from a nozzle follows a parabolic path modeled by the equation y = -0.2(x - 5)² + 10, where y is the height of the water in meters above the basin and x is the horizontal distance in meters from the nozzle. What is the vertex of the parabola, and what does it represent in this context? Answer: (5, 10); the maximum height of the water is 10 meters at a horizontal distance of 5 meters from the nozzle. Solution: The equation is y = -0.2(x - 5)² + 10. This is in vertex form y = a(x - h)² + k, where a = -0.2, h = 5, and k = 10. The vertex is (h, k) = (5, 10).
   Full step-by-step solution

   The equation is y = -0.2(x - 5)² + 10. This is in vertex form y = a(x - h)² + k, where a = -0.2, h = 5, and k = 10. The vertex is (h, k) = (5, 10). Since a = -0.2 is negative, the parabola opens downward, so the vertex is the maximum point. In this context, the vertex means that the water reaches its highest point of 10 meters when it is 5 meters horizontally from the nozzle.

5. A drone is flying along a parabolic path to survey a rectangular field. The drone's height above ground is modeled by the function h(t) = -2t² + 12t + 8, where h is height in meters and t is time in seconds. The drone needs to take a high-resolution photo when it reaches its maximum altitude. At what time should the drone capture the photo? Answer: 3 Solution: We are given the height function: h(t) = -2t² + 12t + 8. This is a quadratic function with a negative coefficient for t², so the graph is a downward-opening parabola.
   Full step-by-step solution

   We are given the height function: h(t) = -2t² + 12t + 8. This is a quadratic function with a negative coefficient for t², so the graph is a downward-opening parabola. The maximum height occurs at the vertex of the parabola. For a quadratic in the form h(t) = at² + bt + c, the t-coordinate of the vertex is given by: t = -b / (2a) Here: a = -2 b = 12 c = 8 Step 1: Write the vertex formula. t = -b / (2a) Step 2: Substitute a and b. t = -12 / (2 * -2) Step 3: Simplify the denominator. 2 * -2 = -4 So t = -12 / (-4) Step 4: Simplify the fraction. Dividing two negatives gives a positive: t = 12 / 4 = 3 Step 5: Interpret the result. The drone reaches maximum height at t = 3 seconds. Thus, the drone should capture the photo at t = 3 seconds. ANSWER: 3

6. A drone is launched from a platform 20 meters high and follows a parabolic path given by the equation h(t) = -2t² + 12t + 20, where h is the height in meters and t is the time in seconds after launch. At what time does the drone reach its maximum height, and what is that maximum height? Answer: 3 seconds, 38 meters Solution: h(t) = -2t² + 12t + 20 This is a quadratic function in the form h(t) = at² + bt + c, where: a = -2 b = 12 c = 20 Since a < 0, the parabola opens downward, so the vertex corresponds to the maximum height.
   Full step-by-step solution

   Let's solve this step by step. We are given the height function: h(t) = -2t² + 12t + 20 --- **Step 1: Identify the type of function** This is a quadratic function in the form h(t) = at² + bt + c, where: a = -2 b = 12 c = 20 Since a < 0, the parabola opens downward, so the vertex corresponds to the maximum height. --- **Step 2: Find the time at which maximum height occurs** For a quadratic function at² + bt + c, the t-coordinate of the vertex is given by: t = -b / (2a) Substitute b = 12 and a = -2: t = -12 / (2 * -2) t = -12 / (-4) t = 3 So the drone reaches its maximum height at t = 3 seconds. --- **Step 3: Find the maximum height** Substitute t = 3 into h(t): h(3) = -2*(3)² + 12*(3) + 20 h(3) = -2*9 + 36 + 20 h(3) = -18 + 36 + 20 h(3) = 18 + 20 h(3) = 38 So the maximum height is 38 meters. --- **Final Answer:** Time of maximum height: 3 seconds Maximum height: 38 meters

7. A parabolic arch bridge has its vertex at point (20, 15) on a coordinate plane where units are in meters. The arch passes through point (0, 0) at ground level. Write the equation of this parabola in vertex form: y = a(x - h)² + k. Answer: y = -3/80(x - 20)² + 15 Solution: Identify the vertex coordinates from the problem: h = 20, k = 15 Substitute into vertex form: y = a(x - 20)² + 15 Use the point (0, 0) to solve for a: 0 = a(0 - 20)² + 15 Simplify: 0 = a(400) + 15 Solve for a: 400a = -15, so a = -15/400 = -3/80 Write the final equation: y = -3/80(x - 20)² + 15
   Full step-by-step solution

   Step 1: Identify the vertex coordinates from the problem: h = 20, k = 15 Step 2: Substitute into vertex form: y = a(x - 20)² + 15 Step 3: Use the point (0, 0) to solve for a: 0 = a(0 - 20)² + 15 Step 4: Simplify: 0 = a(400) + 15 Step 5: Solve for a: 400a = -15, so a = -15/400 = -3/80 Step 6: Write the final equation: y = -3/80(x - 20)² + 15