Graph Key Features

lessonbunny.com

Answer Key & Explanations

Graph Key Features · Grade 9 · Worksheet 2

1. Matiu's function graph shows a parabola with vertex at (4, 15), x-intercepts at (1, 0) and (7, 0), and y-intercept at (0, 14). Identify the intervals where the function is increasing and decreasing. Answer: Increasing: (-∞, 4); Decreasing: (4, ∞) Solution: The vertex is at (4, 15) and the x-intercepts are at (1, 0) and (7, 0). Since the x-intercepts are on either side of the vertex (1 < 4 < 7), the parabola opens downward.
   Full step-by-step solution

   Step 1: The vertex is at (4, 15) and the x-intercepts are at (1, 0) and (7, 0). Step 2: Since the x-intercepts are on either side of the vertex (1 < 4 < 7), the parabola opens downward. Step 3: For a downward-opening parabola, the function increases as it approaches the vertex from the left and decreases after passing the vertex. Step 4: Therefore, the function is increasing on the interval (-∞, 4). Step 5: The function is decreasing on the interval (4, ∞). The answer is: Increasing: (-∞, 4); Decreasing: (4, ∞)

2. A drone is flying over a park, and its height above ground is modeled by the function h(t) = -2t² + 12t + 5, where h is the height in meters and t is the time in seconds after launch. After how many seconds does the drone reach its maximum height? Answer: 3 Solution: We are given the height function: h(t) = -2t² + 12t + 5. This is a quadratic function in the form h(t) = at² + bt + c, where a = -2, b = 12, c = 5.
   Full step-by-step solution

   We are given the height function: h(t) = -2t² + 12t + 5. This is a quadratic function in the form h(t) = at² + bt + c, where a = -2, b = 12, c = 5. Since a is negative (a = -2), the parabola opens downward, so the vertex of the parabola gives the maximum height. 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 after 3 seconds. We can check: h(3) = -2*(3)² + 12*3 + 5 = -2*9 + 36 + 5 = -18 + 36 + 5 = 23 meters maximum height. Thus, the answer is 3 seconds.

3. f(x) = 2x² - 8x + 3; f(5) = ? Answer: 13 Solution: Start with the function f(x) = 2x² - 8x + 3 Substitute x = 5 into the function: f(5) = 2(5)² - 8(5) + 3 Calculate the exponent first: (5)² = 25 Multiply: 2 × 25 = 50 and -8 × 5 = -40 Combine all terms: 50 - 40 + 3 Simplify: 50 - 40 = 10, then 10 + 3 = 13 The answer is 13.
   Full step-by-step solution

   Step 1: Start with the function f(x) = 2x² - 8x + 3 Step 2: Substitute x = 5 into the function: f(5) = 2(5)² - 8(5) + 3 Step 3: Calculate the exponent first: (5)² = 25 Step 4: Multiply: 2 × 25 = 50 and -8 × 5 = -40 Step 5: Combine all terms: 50 - 40 + 3 Step 6: Simplify: 50 - 40 = 10, then 10 + 3 = 13 The answer is 13.

4. Kaia graphs a quadratic function on a coordinate plane. The parabola opens upward and has its vertex at (12, -16). The graph also passes through the point (8, 0). Identify the x-intercepts of the function and state the interval(s) where the function is increasing. Answer: x-intercepts at x = 8 and x = 16; increasing on (12, infinity) Solution: Write the vertex form: y = a(x - h)^2 + k, with vertex (h, k) = (12, -16). So y = a(x - 12)^2 - 16. Substitute the point (8, 0): 0 = a(8 - 12)^2 - 16 = a(-4)^2 - 16 = 16a - 16.
   Full step-by-step solution

   Step 1: Write the vertex form: y = a(x - h)^2 + k, with vertex (h, k) = (12, -16). So y = a(x - 12)^2 - 16. Step 2: Substitute the point (8, 0): 0 = a(8 - 12)^2 - 16 = a(-4)^2 - 16 = 16a - 16. Step 3: Solve for a: 16a = 16 → a = 1. Step 4: The equation is y = (x - 12)^2 - 16. Step 5: To find x-intercepts, set y = 0: 0 = (x - 12)^2 - 16 → (x - 12)^2 = 16 → x - 12 = ±4 → x = 12 ± 4. Step 6: The x-intercepts are at x = 12 - 4 = 8 and x = 12 + 4 = 16. Step 7: Since the parabola opens upward (a = 1 > 0), it decreases to the left of the vertex and increases to the right. The vertex is at x = 12, so the function is increasing on the interval (12, infinity). Final answer: x-intercepts at x = 8 and x = 16; increasing on (12, infinity).

5. Mere's function graph shows a parabola with vertex at (6, 18), x-intercepts at (2, 0) and (10, 0), and y-intercept at (0, 40). Identify the interval where the function is increasing. Answer: (2, 6) Solution: Determine the direction the parabola opens. Since the vertex at (6, 18) is above the x-intercepts at (2, 0) and (10, 0), the parabola opens downward.
   Full step-by-step solution

   Step 1: Determine the direction the parabola opens. Since the vertex at (6, 18) is above the x-intercepts at (2, 0) and (10, 0), the parabola opens downward. Step 2: For a downward-opening parabola, the function increases on the left side of the vertex and decreases on the right side. Step 3: The vertex is at x = 6, so the function increases as x approaches 6 from the left. Step 4: The function starts increasing from the left x-intercept at x = 2 (where the graph crosses the x-axis) up to the vertex at x = 6. Step 5: Therefore, the function is increasing on the interval (2, 6). The answer is (2, 6).

6. f(x) = 2x² - 8x + 6, find f(3) = ? Answer: 0 Solution: We are given the function f(x) = 2x² - 8x + 6, and we need to find f(3). Write down the function. f(x) = 2x² - 8x + 6 Substitute x = 3 into the function.
   Full step-by-step solution

   We are given the function f(x) = 2x² - 8x + 6, and we need to find f(3). Step 1: Write down the function. f(x) = 2x² - 8x + 6 Step 2: Substitute x = 3 into the function. f(3) = 2*(3)² - 8*(3) + 6 Step 3: Calculate the exponent first: (3)² = 9. f(3) = 2*9 - 8*3 + 6 Step 4: Perform the multiplications. 2*9 = 18 8*3 = 24 So f(3) = 18 - 24 + 6 Step 5: Perform the addition and subtraction from left to right. 18 - 24 = -6 -6 + 6 = 0 Step 6: State the final answer. f(3) = 0

7. Isabella is analyzing the graph of f(x) = 3x² - 18x + 24. Identify the vertex coordinates, x-intercepts, and intervals where the function is increasing. Answer: Vertex: (3, -3); x-intercepts: (2, 0) and (4, 0); increasing: (3, ∞) Solution: Find the vertex coordinates using x = -b/(2a). For f(x) = 3x² - 18x + 24, a = 3, b = -18, c = 24. x = -(-18)/(2×3) = 18/6 = 3.
   Full step-by-step solution

   Step 1: Find the vertex coordinates using x = -b/(2a). For f(x) = 3x² - 18x + 24, a = 3, b = -18, c = 24. x = -(-18)/(2×3) = 18/6 = 3. Then find f(3) = 3(3)² - 18(3) + 24 = 3(9) - 54 + 24 = 27 - 54 + 24 = -3. So vertex is (3, -3). Step 2: Find x-intercepts by solving 3x² - 18x + 24 = 0. Divide by 3: x² - 6x + 8 = 0. Factor: (x - 2)(x - 4) = 0. So x = 2 and x = 4. x-intercepts are (2, 0) and (4, 0). Step 3: Determine intervals of increase/decrease. Since a = 3 > 0, the parabola opens upward. The function decreases on (-∞, 3) and increases on (3, ∞). Final answer: Vertex: (3, -3); x-intercepts: (2, 0) and (4, 0); increasing: (3, ∞)

8. Kai is studying the population P(t) = 60 * 2^(t/2) of bacteria in a lab experiment, where t is hours. From the graph of this function, identify the y-intercept and describe the interval(s) where the population is increasing or decreasing. Answer: 60 Solution: The function P(t) = 60 * 2^(t/2) is an exponential growth function. The y-intercept occurs when t = 0: P(0) = 60 * 2^(0) = 60 * 1 = 60. Since the base 2 > 1, the function is always increasing for all t ≥ 0.
   Full step-by-step solution

   Step 1: The function P(t) = 60 * 2^(t/2) is an exponential growth function. Step 2: The y-intercept occurs when t = 0: P(0) = 60 * 2^(0) = 60 * 1 = 60. Step 3: Since the base 2 > 1, the function is always increasing for all t ≥ 0. Step 4: There are no decreasing intervals; the population grows continuously. Answer: The y-intercept is 60 (initial population). The population is increasing on the interval (0, ∞).

9. Mason is analyzing the graph of f(x) = -2x² + 12x - 7. Identify the vertex coordinates, the y-intercept, and the interval where the function is increasing. Answer: Vertex: (3, 11), y-intercept: (0, -7), increasing interval: (-∞, 3) Solution: Identify a, b, c from f(x) = -2x² + 12x - 7. Here a = -2, b = 12, c = -7. Find the x-coordinate of the vertex using x = -b/(2a) = -12/(2*(-2)) = -12/(-4) = 3.
   Full step-by-step solution

   Step 1: Identify a, b, c from f(x) = -2x² + 12x - 7. Here a = -2, b = 12, c = -7. Step 2: Find the x-coordinate of the vertex using x = -b/(2a) = -12/(2*(-2)) = -12/(-4) = 3. Step 3: Find the y-coordinate by substituting x = 3 into the function: f(3) = -2(3)² + 12(3) - 7 = -2(9) + 36 - 7 = -18 + 36 - 7 = 11. So the vertex is (3, 11). Step 4: Find the y-intercept by setting x = 0: f(0) = -2(0)² + 12(0) - 7 = -7. So the y-intercept is (0, -7). Step 5: Since a = -2 is negative, the parabola opens downward. For a downward-opening parabola, the function increases on the left side of the vertex and decreases on the right side. The vertex x-coordinate is 3, so the function is increasing on the interval (-∞, 3). The answer is: Vertex: (3, 11), y-intercept: (0, -7), increasing interval: (-∞, 3).