Graph Key Features

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Answer Key & Explanations

Graph Key Features · Grade 9 · Worksheet 1

1. f(x) = -2x² + 12x - 10; vertex = ? Answer: (3, 8) Solution: Identify the coefficients from the quadratic function f(x) = ax² + bx + c. Here, a = -2, b = 12, c = -10. The x-coordinate of the vertex is given by the formula x = -b/(2a).
   Full step-by-step solution

   Step 1: Identify the coefficients from the quadratic function f(x) = ax² + bx + c. Here, a = -2, b = 12, c = -10. Step 2: The x-coordinate of the vertex is given by the formula x = -b/(2a). Step 3: Substitute the values: x = -12 / (2 * -2) = -12 / -4 = 3. Step 4: Substitute x = 3 back into the original function to find the y-coordinate: f(3) = -2(3)² + 12(3) - 10. Step 5: Calculate f(3): -2(9) + 36 - 10 = -18 + 36 - 10 = 8. Step 6: The vertex is the point (x, y), so the vertex is (3, 8).

2. f(x) = 3x² - 12x + 5; f(4) = ? Answer: 5 Solution: Write the function: f(x) = 3x² - 12x + 5 Substitute x = 4: f(4) = 3(4)² - 12(4) + 5 Calculate the exponent: (4)² = 16 Multiply: 3 × 16 = 48 and 12 × 4 = 48 Substitute back: f(4) = 48 - 48 + 5 Simplify: 48 - 48 = 0, then 0 + 5 = 5 The answer is 5.
   Full step-by-step solution

   Step 1: Write the function: f(x) = 3x² - 12x + 5 Step 2: Substitute x = 4: f(4) = 3(4)² - 12(4) + 5 Step 3: Calculate the exponent: (4)² = 16 Step 4: Multiply: 3 × 16 = 48 and 12 × 4 = 48 Step 5: Substitute back: f(4) = 48 - 48 + 5 Step 6: Simplify: 48 - 48 = 0, then 0 + 5 = 5 The answer is 5.

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

   We are given the height function: h(t) = -2t^2 + 12t + 5. This is a quadratic function in the form h(t) = a t^2 + b t + 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. The vertex occurs at time t = -b / (2a). Step 1: Identify a and b. a = -2 b = 12 Step 2: Plug into the vertex formula. t = -b / (2a) = -12 / (2 * -2) Step 3: Simplify the denominator. 2 * a = 2 * (-2) = -4 Step 4: Simplify the fraction. t = -12 / (-4) = 12 / 4 = 3 So, the drone reaches its maximum height at t = 3 seconds. Final answer: 3

4. Isabella is analyzing the graph of f(x) = 2x² - 16x + 30. The graph has x-intercepts at (3, 0) and (5, 0), a y-intercept at (0, 30), and a vertex at (4, -2). Over what interval is the function decreasing? Answer: (-∞, 4) Solution: Identify the direction the parabola opens. Since the coefficient of x² is positive (2), the parabola opens upward.
   Full step-by-step solution

   Step 1: Identify the direction the parabola opens. Since the coefficient of x² is positive (2), the parabola opens upward. Step 2: For an upward-opening parabola, the function decreases on the left side of the vertex and increases on the right side of the vertex. Step 3: The vertex is at (4, -2), so the x-coordinate of the vertex is 4. Step 4: The function decreases as x approaches 4 from the left, so the decreasing interval is all x-values less than 4. Step 5: Write the interval in proper notation: (-∞, 4) The function is decreasing on the interval (-∞, 4).

5. Mere is studying the population P(t) = 150 * 2^(t/3) 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: 150 Solution: The function P(t) = 150 * 2^(t/3) is an exponential growth function. The y-intercept occurs when t = 0: P(0) = 150 * 2^(0) = 150 * 1 = 150. 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) = 150 * 2^(t/3) is an exponential growth function. Step 2: The y-intercept occurs when t = 0: P(0) = 150 * 2^(0) = 150 * 1 = 150. 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 150 (initial population). The population is increasing on the interval (0, ∞).

6. A drone is launched from a platform 15 meters high. Its height above ground is modeled by the function h(t) = -2t² + 8t + 15, where t is time in seconds and h(t) is height in meters. After how many seconds will the drone reach its maximum height? Answer: 2 Solution: We are given the height function: h(t) = -2t² + 8t + 15. This is a quadratic function in the form h(t) = at² + bt + c, where a = -2, b = 8, c = 15.
   Full step-by-step solution

   We are given the height function: h(t) = -2t² + 8t + 15. This is a quadratic function in the form h(t) = at² + bt + c, where a = -2, b = 8, c = 15. Since a is negative (a = -2), the parabola opens downward, so the vertex of the parabola gives the maximum height. For a quadratic function in the form at² + bt + c, the t-coordinate of the vertex (which gives the time at maximum height) is found by: t = -b / (2a) Step 1: Identify a and b. a = -2 b = 8 Step 2: Substitute into the vertex formula. t = -b / (2a) = -8 / (2 * (-2)) Step 3: Simplify the denominator. 2 * (-2) = -4 Step 4: Simplify the fraction. t = -8 / (-4) = 8 / 4 = 2 So, the drone reaches its maximum height at t = 2 seconds. Final answer: 2

7. Emma's quadratic function graph has x-intercepts at (-5, 0) and (5, 0), a y-intercept at (0, -25), and vertex at (0, -25). What is the maximum value of the function? Answer: -25 Solution: Identify the vertex coordinates from the problem: (0, -25) For a quadratic function, the maximum or minimum value is the y-coordinate of the vertex Since the parabola opens downward (indicated by the vertex being the highest point), the y-coordinate -25 represents the maximum value The maximum…
   Full step-by-step solution

   Step 1: Identify the vertex coordinates from the problem: (0, -25) Step 2: For a quadratic function, the maximum or minimum value is the y-coordinate of the vertex Step 3: Since the parabola opens downward (indicated by the vertex being the highest point), the y-coordinate -25 represents the maximum value Step 4: The maximum value of the function is -25

8. Nikau is studying the population P(t) = 290 * 2^(t/3) 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: 290 Solution: The function P(t) = 290 * 2^(t/3) is an exponential growth function. The y-intercept occurs when t = 0: P(0) = 290 * 2^(0) = 290 * 1 = 290. 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) = 290 * 2^(t/3) is an exponential growth function. Step 2: The y-intercept occurs when t = 0: P(0) = 290 * 2^(0) = 290 * 1 = 290. 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 290 (initial population). The population is increasing on the interval (0, ∞).

9. f(x) = 2x² - 5x + 1; f(3) = ? Answer: 4 Solution: Write the function: f(x) = 2x² - 5x + 1 Substitute x = 3 into the function: f(3) = 2(3)² - 5(3) + 1 Calculate the exponent first: 3² = 9 Multiply: 2 × 9 = 18 and 5 × 3 = 15 Rewrite the expression: f(3) = 18 - 15 + 1 Perform addition and subtraction from left to right: 18 - 15 = 3, then 3 + 1 = 4…
   Full step-by-step solution

   Step 1: Write the function: f(x) = 2x² - 5x + 1 Step 2: Substitute x = 3 into the function: f(3) = 2(3)² - 5(3) + 1 Step 3: Calculate the exponent first: 3² = 9 Step 4: Multiply: 2 × 9 = 18 and 5 × 3 = 15 Step 5: Rewrite the expression: f(3) = 18 - 15 + 1 Step 6: Perform addition and subtraction from left to right: 18 - 15 = 3, then 3 + 1 = 4 The answer is 4.