Graph Key Features

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Graph Key Features · Grade 9 · Worksheet 2

1. The population of a certain bacteria colony is modeled by the function P(t) = 1200 * 2^(t/3), where t is the time in hours and P(t) is the population size. After how many hours will the bacteria population reach 19,200? Answer: 12 Solution: Set up the equation: 1200 * 2^(t/3) = 19200 Divide both sides by 1200: 2^(t/3) = 19200/1200 Simplify: 2^(t/3) = 16 Recognize that 16 = 2^4, so: 2^(t/3) = 2^4 Since the bases are equal, set the exponents equal: t/3 = 4 Multiply both sides by 3: t = 4 * 3 Calculate: t = 12 The answer is 12 hours.
   Full step-by-step solution

   Step 1: Set up the equation: 1200 * 2^(t/3) = 19200 Step 2: Divide both sides by 1200: 2^(t/3) = 19200/1200 Step 3: Simplify: 2^(t/3) = 16 Step 4: Recognize that 16 = 2^4, so: 2^(t/3) = 2^4 Step 5: Since the bases are equal, set the exponents equal: t/3 = 4 Step 6: Multiply both sides by 3: t = 4 * 3 Step 7: Calculate: t = 12 The answer is 12 hours.

2. Emma is analyzing the graph of a quadratic function. The graph has x-intercepts at (-5, 0) and (5, 0), a y-intercept at (0, -25), and a vertex at (0, -25). Over what interval is the function increasing? Answer: x > 0 Solution: The vertex is at (0, -25), which is also the y-intercept. Since the parabola opens upward (x-intercepts are at -5 and 5, vertex is below them), the function decreases until the vertex and increases after the vertex.
   Full step-by-step solution

   Step 1: The vertex is at (0, -25), which is also the y-intercept. Step 2: Since the parabola opens upward (x-intercepts are at -5 and 5, vertex is below them), the function decreases until the vertex and increases after the vertex. Step 3: The vertex occurs at x = 0, so the function is increasing for all x-values greater than 0. Step 4: Therefore, the function is increasing on the interval x > 0.

3. f(x) = 2x³ - 5x² + 3x - 7; f(-2) = ? Answer: -49 Solution: Substitute x = -2 into the function: f(-2) = 2(-2)³ - 5(-2)² + 3(-2) - 7 Calculate (-2)³ = (-2) × (-2) × (-2) = -8 Calculate (-2)² = (-2) × (-2) = 4 Substitute these values: f(-2) = 2(-8) - 5(4) + 3(-2) - 7 Multiply: 2(-8) = -16, -5(4) = -20, 3(-2) = -6 Combine all terms: -16 - 20 - 6 - 7…
   Full step-by-step solution

   Step 1: Substitute x = -2 into the function: f(-2) = 2(-2)³ - 5(-2)² + 3(-2) - 7 Step 2: Calculate (-2)³ = (-2) × (-2) × (-2) = -8 Step 3: Calculate (-2)² = (-2) × (-2) = 4 Step 4: Substitute these values: f(-2) = 2(-8) - 5(4) + 3(-2) - 7 Step 5: Multiply: 2(-8) = -16, -5(4) = -20, 3(-2) = -6 Step 6: Combine all terms: -16 - 20 - 6 - 7 Step 7: Calculate step by step: -16 - 20 = -36, -36 - 6 = -42, -42 - 7 = -49 The answer is -49.

4. 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, ∞)

5. A quadratic function is graphed on a coordinate plane. The parabola opens downward and has its vertex at (2, 9). It passes through the point (5, 0). What is the equation of this quadratic function in standard form? Answer: y = -x² + 4x + 5 Solution: Since the parabola opens downward and has vertex (h, k) = (2, 9), we use the vertex form: y = a(x - h)^2 + k y = a(x - 2)^2 + 9 Use the given point (5, 0) to find 'a' The point (5, 0) means x = 5, y = 0.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Identify the vertex form of a quadratic function** Since the parabola opens downward and has vertex (h, k) = (2, 9), we use the vertex form: y = a(x - h)^2 + k y = a(x - 2)^2 + 9 --- **Step 2: Use the given point (5, 0) to find 'a'** The point (5, 0) means x = 5, y = 0. Substitute into the equation: 0 = a(5 - 2)^2 + 9 0 = a(3)^2 + 9 0 = 9a + 9 --- **Step 3: Solve for a** 9a + 9 = 0 9a = -9 a = -1 --- **Step 4: Write the vertex form equation** y = -1(x - 2)^2 + 9 --- **Step 5: Expand to standard form y = ax^2 + bx + c** First expand (x - 2)^2: (x - 2)^2 = x^2 - 4x + 4 Multiply by -1: -1(x^2 - 4x + 4) = -x^2 + 4x - 4 Now add 9: y = -x^2 + 4x - 4 + 9 y = -x^2 + 4x + 5 --- **Step 6: Verify with the given point** Vertex: (2, 9) y = -4 + 8 + 5 = 9 ✓ Point (5, 0): y = -25 + 20 + 5 = 0 ✓ --- **Final answer:** y = -x^2 + 4x + 5

6. Isabella is analyzing the graph of f(x) = 3x² - 18x + 15. Identify the vertex coordinates, x-intercepts, and intervals where the function is increasing. Answer: Vertex: (3, -12); x-intercepts: (1, 0) and (5, 0); increasing: (3, ∞) Solution: Find the vertex using x = -b/(2a) where a = 3, b = -18 x = -(-18)/(2*3) = 18/6 = 3 Find the y-coordinate by substituting x = 3 into f(x) f(3) = 3(3)² - 18(3) + 15 = 3(9) - 54 + 15 = 27 - 54 + 15 = -12 Vertex: (3, -12) Find x-intercepts by setting f(x) = 0 3x² - 18x + 15 = 0 Divide by 3: x² - 6x…
   Full step-by-step solution

   Step 1: Find the vertex using x = -b/(2a) where a = 3, b = -18 x = -(-18)/(2*3) = 18/6 = 3 Step 2: Find the y-coordinate by substituting x = 3 into f(x) f(3) = 3(3)² - 18(3) + 15 = 3(9) - 54 + 15 = 27 - 54 + 15 = -12 Vertex: (3, -12) Step 3: Find x-intercepts by setting f(x) = 0 3x² - 18x + 15 = 0 Divide by 3: x² - 6x + 5 = 0 Factor: (x - 1)(x - 5) = 0 x = 1 or x = 5 x-intercepts: (1, 0) and (5, 0) Step 4: Determine intervals of increase/decrease Since a = 3 > 0, the parabola opens upward Function decreases on (-∞, 3) and increases on (3, ∞)

7. 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

8. Emma is designing a rectangular garden with a fixed perimeter of 100 meters. She wants to maximize the area of the garden. If the length of the garden is represented by x meters, write the area function A(x) in standard quadratic form and determine the dimensions that give the maximum area. Answer: 25 by 25 Solution: Let the length be x meters and the width be w meters. The perimeter is 2x + 2w = 100, so x + w = 50, which means w = 50 - x. The area A = length × width = x(50 - x) = 50x - x².
   Full step-by-step solution

   Step 1: Let the length be x meters and the width be w meters. Step 2: The perimeter is 2x + 2w = 100, so x + w = 50, which means w = 50 - x. Step 3: The area A = length × width = x(50 - x) = 50x - x². Step 4: The area function is A(x) = -x² + 50x, which is a quadratic function opening downward. Step 5: The maximum occurs at the vertex x = -b/(2a) = -50/(2×(-1)) = 25. Step 6: When x = 25, w = 50 - 25 = 25. Step 7: The dimensions that maximize the area are 25 meters by 25 meters.

9. A quadratic function is graphed on a coordinate plane. The parabola opens downward and has its vertex at (2, 9). The parabola intersects the x-axis at two points: one at (-1, 0) and another at (5, 0). What is the equation of this quadratic function in standard form? Answer: y = -x² + 4x + 5 Solution: The parabola opens downward, so the coefficient of \( x^2 \) is negative. We know the vertex is at (2, 9), so we can use the vertex form: y = a(x - h)^2 + k where \( h = 2 \), \( k = 9 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the form of the quadratic equation** The parabola opens downward, so the coefficient of \( x^2 \) is negative. We know the vertex is at (2, 9), so we can use the vertex form: \[ y = a(x - h)^2 + k \] where \( h = 2 \), \( k = 9 \). So: \[ y = a(x - 2)^2 + 9 \] --- **Step 2: Use one x-intercept to find \( a \)** The parabola passes through (-1, 0). Substitute \( x = -1 \), \( y = 0 \) into the equation: \[ 0 = a(-1 - 2)^2 + 9 \] \[ 0 = a(-3)^2 + 9 \] \[ 0 = 9a + 9 \] \[ 9a = -9 \] \[ a = -1 \] --- **Step 3: Write the vertex form with \( a \)** \[ y = -1(x - 2)^2 + 9 \] --- **Step 4: Expand to standard form \( y = ax^2 + bx + c \)** First expand \( (x - 2)^2 \): \[ (x - 2)^2 = x^2 - 4x + 4 \] Multiply by -1: \[ -1(x^2 - 4x + 4) = -x^2 + 4x - 4 \] Now add 9: \[ y = -x^2 + 4x - 4 + 9 \] \[ y = -x^2 + 4x + 5 \] --- **Step 5: Verify with the other x-intercept** Check \( x = 5 \): \[ y = -(5)^2 + 4(5) + 5 = -25 + 20 + 5 = 0 \] Yes, it passes through (5, 0). --- **Final answer:** \[ y = -x^2 + 4x + 5 \]