Quadratic Vertex Form

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Quadratic Vertex Form · Grade 10 · Worksheet 3

1. A drone is launched from a platform 12 meters high. Its height above ground is modeled by the function h(t) = -2t² + 8t + 12, where t is time in seconds. After how many seconds will the drone reach its maximum height? Answer: 2 Solution: We are given the height function: h(t) = -2t² + 8t + 12. This is a quadratic function in the form h(t) = at² + bt + c, with a = -2, b = 8, c = 12.
   Full step-by-step solution

   We are given the height function: h(t) = -2t² + 8t + 12. This is a quadratic function in the form h(t) = at² + bt + c, with a = -2, b = 8, c = 12. Since a < 0, the parabola opens downward, so the vertex gives the maximum height. The vertex occurs at t = -b / (2a). Step 1: Identify a and b. a = -2, b = 8. Step 2: Apply the vertex formula. t = -b / (2a) t = -8 / (2 * (-2)) Step 3: Simplify the denominator. 2 * (-2) = -4 So t = -8 / (-4) Step 4: Simplify the fraction. Negative divided by negative is positive: t = 8 / 4 t = 2 Step 5: Interpret the result. The drone reaches its maximum height at t = 2 seconds. Final answer: 2

2. Convert f(x) = 4x² - 24x + 35 to vertex form f(x) = a(x - h)² + k = ? Answer: 4(x - 3)² - 1 Solution: Factor out the leading coefficient from the first two terms: f(x) = 4(x² - 6x) + 35 Complete the square inside the parentheses: Take half of -6, which is -3, and square it to get 9 Add and subtract 9 inside the parentheses: f(x) = 4(x² - 6x + 9 - 9) + 35 Rewrite as: f(x) = 4[(x² - 6x + 9) - 9] +…
   Full step-by-step solution

   Step 1: Factor out the leading coefficient from the first two terms: f(x) = 4(x² - 6x) + 35 Step 2: Complete the square inside the parentheses: Take half of -6, which is -3, and square it to get 9 Step 3: Add and subtract 9 inside the parentheses: f(x) = 4(x² - 6x + 9 - 9) + 35 Step 4: Rewrite as: f(x) = 4[(x² - 6x + 9) - 9] + 35 Step 5: Factor the perfect square trinomial: f(x) = 4[(x - 3)² - 9] + 35 Step 6: Distribute the 4: f(x) = 4(x - 3)² - 36 + 35 Step 7: Combine constants: f(x) = 4(x - 3)² - 1 The vertex form is 4(x - 3)² - 1.

3. Convert f(x) = -4x² + 24x - 35 to vertex form f(x) = a(x - h)² + k = ? Answer: -4(x - 3)² + 1 Solution: Factor out -4 from the first two terms: f(x) = -4(x² - 6x) - 35 Complete the square inside the parentheses: Take half of -6, which is -3, and square it to get 9 Add and subtract 9 inside the parentheses: f(x) = -4(x² - 6x + 9 - 9) - 35 Rewrite as: f(x) = -4[(x² - 6x + 9) - 9] - 35 Simplify: f(x)…
   Full step-by-step solution

   Step 1: Factor out -4 from the first two terms: f(x) = -4(x² - 6x) - 35 Step 2: Complete the square inside the parentheses: Take half of -6, which is -3, and square it to get 9 Step 3: Add and subtract 9 inside the parentheses: f(x) = -4(x² - 6x + 9 - 9) - 35 Step 4: Rewrite as: f(x) = -4[(x² - 6x + 9) - 9] - 35 Step 5: Simplify: f(x) = -4[(x - 3)² - 9] - 35 Step 6: Distribute the -4: f(x) = -4(x - 3)² + 36 - 35 Step 7: Combine constants: f(x) = -4(x - 3)² + 1 The vertex form is -4(x - 3)² + 1

4. Mason is analyzing the graph of a parabola in vertex form. The parabola opens downward, and its vertex is located at (9, 12). The graph also passes through the point (13, 4). Write the equation of this parabola in vertex form: y = a(x - h)² + k. Answer: y = -1/2(x - 9)² + 12 Solution: Identify the vertex. The vertex is at (9, 12), so h = 9 and k = 12. Step 2: Write the partial equation: y = a(x - 9)² + 12.
   Full step-by-step solution

   Step 1: Identify the vertex. The vertex is at (9, 12), so h = 9 and k = 12. Step 2: Write the partial equation: y = a(x - 9)² + 12. Step 3: Substitute the given point (13, 4) into the equation: 4 = a(13 - 9)² + 12. Step 4: Simplify: 4 = a(4)² + 12, so 4 = 16a + 12. Step 5: Subtract 12 from both sides: 4 - 12 = 16a, so -8 = 16a. Step 6: Divide both sides by 16: a = -8/16 = -1/2. Step 7: Write the final equation: y = -1/2(x - 9)² + 12. The answer is y = -1/2(x - 9)² + 12.

5. Noah is designing a water fountain that shoots a stream of water into the air. The height of the water stream, in meters above the fountain's base, is modeled by the function h(t) = -2(t - 1)² + 6, where t is time in seconds after the water leaves the nozzle. Noah wants to know the maximum height the water reaches and the time at which it occurs. What are these two values? Answer: t = 1 second, h = 6 meters Solution: The function is given in vertex form: h(t) = -2(t - 1)² + 6. In vertex form, y = a(x - h)² + k, the vertex is at (h, k). Here, h = 1 and k = 6.
   Full step-by-step solution

   Step 1: The function is given in vertex form: h(t) = -2(t - 1)² + 6. Step 2: In vertex form, y = a(x - h)² + k, the vertex is at (h, k). Here, h = 1 and k = 6. Step 3: Since a = -2 is negative, the parabola opens downward, so the vertex represents the maximum point. Step 4: The x-coordinate (h) is the time when the maximum height occurs: t = 1 second. Step 5: The y-coordinate (k) is the maximum height: h = 6 meters. The answer is t = 1 second and h = 6 meters.

6. Convert f(x) = 2x² + 12x - 8 to vertex form f(x) = a(x - h)² + k = ? Answer: 2(x + 3)² - 26 Solution: Factor out the coefficient of x² from the first two terms: f(x) = 2(x² + 6x) - 8 Complete the square inside the parentheses: Take half of 6 (which is 3), square it (9), and add and subtract it inside: f(x) = 2(x² + 6x + 9 - 9) - 8 Rewrite as a perfect square and simplify: f(x) = 2[(x + 3)² - 9]…
   Full step-by-step solution

   Step 1: Factor out the coefficient of x² from the first two terms: f(x) = 2(x² + 6x) - 8 Step 2: Complete the square inside the parentheses: Take half of 6 (which is 3), square it (9), and add and subtract it inside: f(x) = 2(x² + 6x + 9 - 9) - 8 Step 3: Rewrite as a perfect square and simplify: f(x) = 2[(x + 3)² - 9] - 8 Step 4: Distribute the 2: f(x) = 2(x + 3)² - 18 - 8 Step 5: Combine constants: f(x) = 2(x + 3)² - 26 The vertex form is f(x) = 2(x + 3)² - 26

7. Convert the quadratic function f(x) = 2x² - 12x + 16 to vertex form f(x) = a(x - h)² + k Answer: 2(x - 3)² - 2 Solution: To convert the quadratic function f(x) = 2x² - 12x + 16 to vertex form f(x) = a(x - h)² + k, we use the method of completing the square. Factor out the coefficient of x² from the first two terms.
   Full step-by-step solution

   To convert the quadratic function f(x) = 2x² - 12x + 16 to vertex form f(x) = a(x - h)² + k, we use the method of completing the square. Step 1: Factor out the coefficient of x² from the first two terms. The coefficient of x² is 2, so we factor 2 out of the terms 2x² and -12x. f(x) = 2(x² - 6x) + 16 Step 2: Complete the square inside the parentheses. Look at the expression inside: x² - 6x. Take the coefficient of x, which is -6. Divide it by 2: -6 / 2 = -3. Square this result: (-3)² = 9. Now, add and subtract this square (9) inside the parentheses. f(x) = 2(x² - 6x + 9 - 9) + 16 Step 3: Group the perfect square trinomial and deal with the constant. The expression x² - 6x + 9 is a perfect square trinomial. It factors as (x - 3)². The -9 remains inside the parentheses, which is being multiplied by 2. f(x) = 2[(x² - 6x + 9) - 9] + 16 f(x) = 2[(x - 3)² - 9] + 16 Step 4: Distribute the 2 and simplify. Distribute the 2 to both terms inside the brackets. f(x) = 2(x - 3)² - 18 + 16 Now combine the constant terms: -18 + 16 = -2. f(x) = 2(x - 3)² - 2 This is the vertex form, where a = 2, h = 3, and k = -2. Final Answer: 2(x - 3)² - 2

8. Mere is analyzing a parabolic water fountain. On a coordinate grid where 1 unit equals 1 meter, the fountain's water stream forms a parabola with its vertex at the point (12, 18). The stream of water lands on the ground at the point (24, 0). Write the equation of this parabola in vertex form: y = a(x - h)² + k. Answer: y = -1/8(x - 12)² + 18 Solution: Identify the vertex coordinates from the problem: h = 12, k = 18. Substitute into vertex form: y = a(x - 12)² + 18. Use the point (24, 0) to solve for a: 0 = a(24 - 12)² + 18.
   Full step-by-step solution

   Step 1: Identify the vertex coordinates from the problem: h = 12, k = 18. Step 2: Substitute into vertex form: y = a(x - 12)² + 18. Step 3: Use the point (24, 0) to solve for a: 0 = a(24 - 12)² + 18. Step 4: Simplify inside the parentheses: 0 = a(12)² + 18. Step 5: Calculate the square: 0 = a(144) + 18. Step 6: Subtract 18 from both sides: -18 = 144a. Step 7: Divide both sides by 144: a = -18/144 = -1/8. Step 8: Write the final equation: y = -1/8(x - 12)² + 18. The answer is y = -1/8(x - 12)² + 18.