Graph Key Features

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Answer Key & Explanations

Graph Key Features · Grade 9 · Worksheet 3

1. Mere's function graph shows a parabola with vertex at (4, 8), x-intercepts at (2, 0) and (6, 0), and y-intercept at (0, 24). Identify the intervals where the function is increasing and decreasing. Answer: Increasing: (-∞, 4); Decreasing: (4, ∞) Solution: The vertex is at (4, 8), which is the highest or lowest point on the parabola.
   Full step-by-step solution

   Step 1: The vertex is at (4, 8), which is the highest or lowest point on the parabola. Step 2: Since the y-intercept (0, 24) is above the vertex (4, 8), and the parabola passes through points (0, 24), (2, 0), (4, 8), and (6, 0), 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) and decreasing on the interval (4, ∞).

2. Emma is analyzing the graph of f(x) = -3x² + 27x - 45. The graph has x-intercepts at (3, 0) and (5, 0), a y-intercept at (0, -45), and a vertex at (4.5, 15.75). Over what interval is the function increasing? Answer: (-∞, 4.5) Solution: Identify the direction the parabola opens. The coefficient of x² is -3 (negative), so the parabola opens downward.
   Full step-by-step solution

   Step 1: Identify the direction the parabola opens. The coefficient of x² is -3 (negative), so 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 (4.5, 15.75), so the x-coordinate of the vertex is 4.5. Step 4: The function increases as x approaches 4.5 from the left, so the increasing interval is all x-values less than 4.5. Step 5: Write the interval in proper notation: (-∞, 4.5) The function is increasing on the interval (-∞, 4.5).

3. A drone is launched from a platform 20 meters high. Its height above ground is modeled by the function h(t) = -5t² + 15t + 20, where t is time in seconds and h(t) is height in meters. After how many seconds will the drone hit the ground? Answer: 4 Solution: We are given the height function: h(t) = -5t² + 15t + 20. The drone hits the ground when h(t) = 0. Set up the equation for h(t) = 0.
   Full step-by-step solution

   We are given the height function: h(t) = -5t² + 15t + 20. The drone hits the ground when h(t) = 0. Step 1: Set up the equation for h(t) = 0. -5t² + 15t + 20 = 0 Step 2: Multiply the entire equation by -1 to make the leading coefficient positive. 5t² - 15t - 20 = 0 Step 3: Divide the entire equation by 5 to simplify. t² - 3t - 4 = 0 Step 4: Factor the quadratic equation. We need two numbers that multiply to -4 and add to -3. Those numbers are -4 and +1. So: (t - 4)(t + 1) = 0 Step 5: Solve for t. t - 4 = 0 → t = 4 t + 1 = 0 → t = -1 Step 6: Interpret the solutions. t = -1 is not meaningful in this context because time cannot be negative. t = 4 is the time when the drone hits the ground. Final answer: 4 seconds.

4. 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.

5. A quadratic function is graphed on a coordinate plane. The parabola opens upward and has its vertex at (-3, -4). The parabola also passes through the point (1, 12). What is the equation of this quadratic function in standard form (ax² + bx + c)? Answer: x² + 6x + 5 Solution: Start with the vertex form of a quadratic: y = a(x - h)² + k, where (h, k) is the vertex.
   Full step-by-step solution

   Step 1: Start with the vertex form of a quadratic: y = a(x - h)² + k, where (h, k) is the vertex. Step 2: Substitute the vertex (-3, -4): y = a(x - (-3))² + (-4) = a(x + 3)² - 4 Step 3: Substitute the point (1, 12) to find 'a': 12 = a(1 + 3)² - 4 Step 4: Simplify: 12 = a(4)² - 4 = 16a - 4 Step 5: Solve for a: 12 + 4 = 16a → 16 = 16a → a = 1 Step 6: Write the equation with a = 1: y = (x + 3)² - 4 Step 7: Expand to standard form: y = (x² + 6x + 9) - 4 = x² + 6x + 5 The answer is x² + 6x + 5.

6. 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.

7. 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).

8. Emma is analyzing the growth of bacteria in a lab experiment. The population P(t) after t hours is modeled by the exponential function P(t) = 500 * 2^(t/3). She needs to determine how long it will take for the bacteria population to reach 4000. How many hours does it take for the population to reach 4000? Answer: 9 Solution: Set up the equation using the given function: 500 * 2^(t/3) = 4000 Divide both sides by 500: 2^(t/3) = 8 Recognize that 8 = 2^3, so: 2^(t/3) = 2^3 Since the bases are equal, set the exponents equal: t/3 = 3 Multiply both sides by 3: t = 9 The bacteria population reaches 4000 after 9 hours.
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 500 * 2^(t/3) = 4000 Step 2: Divide both sides by 500: 2^(t/3) = 8 Step 3: Recognize that 8 = 2^3, so: 2^(t/3) = 2^3 Step 4: Since the bases are equal, set the exponents equal: t/3 = 3 Step 5: Multiply both sides by 3: t = 9 Step 6: The bacteria population reaches 4000 after 9 hours.

9. A cubic function is graphed on a coordinate plane. The curve passes through the points (-2, 0), (1, 0), and (3, 0), and also passes through the point (0, 6). What is the equation of this cubic function in standard form? Answer: y = -x³ + 2x² + 5x + 6 Solution: Since the graph passes through (-2, 0), (1, 0), and (3, 0), these are the x-intercepts, so the factors are (x + 2), (x - 1), and (x - 3). The general form is y = a(x + 2)(x - 1)(x - 3). Use the point (0, 6) to find 'a'.
   Full step-by-step solution

   Step 1: Since the graph passes through (-2, 0), (1, 0), and (3, 0), these are the x-intercepts, so the factors are (x + 2), (x - 1), and (x - 3). Step 2: The general form is y = a(x + 2)(x - 1)(x - 3). Step 3: Use the point (0, 6) to find 'a'. Substitute x = 0 and y = 6: 6 = a(0 + 2)(0 - 1)(0 - 3) Step 4: Simplify: 6 = a(2)(-1)(-3) = a(6) Step 5: Solve for a: 6 = 6a, so a = 1 Step 6: The equation is y = 1(x + 2)(x - 1)(x - 3) Step 7: Expand the factors: First multiply (x + 2)(x - 1) = x² + x - 2 Step 8: Then multiply (x² + x - 2)(x - 3) = x³ - 3x² + x² - 3x - 2x + 6 = x³ - 2x² - 5x + 6 Step 9: The final equation is y = x³ - 2x² - 5x + 6