Equivalent Forms

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Equivalent Forms · Grade 9 · Worksheet 3

1. Ava draws a parabola that passes through the points (7, 0) and (13, 0) and has its vertex at (10, -9). Which of the following equivalent forms of the quadratic function best reveals the y-intercept of the parabola, and what is the y-intercept? Form A: y = (x - 10)^2 - 9 Form B: y = (x - 7)(x - 13) Form C: y = x^2 - 20x + 91 Answer: 91 Solution: Understand what the y-intercept represents. The y-intercept is the point where x = 0, so we need to find f(0) for the quadratic function. - Form A: y = (x - 10)^2 - 9.
   Full step-by-step solution

   Step 1: Understand what the y-intercept represents. The y-intercept is the point where x = 0, so we need to find f(0) for the quadratic function. Step 2: Examine each form: - Form A: y = (x - 10)^2 - 9. To find f(0), we substitute: (0 - 10)^2 - 9 = 100 - 9 = 91. This works but requires a calculation. - Form B: y = (x - 7)(x - 13). To find f(0), substitute: (0 - 7)(0 - 13) = (-7)(-13) = 91. This also works and is a simple multiplication. - Form C: y = x^2 - 20x + 91. To find f(0), substitute: 0^2 - 20(0) + 91 = 91. This is the simplest because the constant term directly gives the y-intercept. Step 3: Form C, the standard form y = ax^2 + bx + c, reveals the y-intercept as the constant term c. Here c = 91, so the y-intercept is 91. Step 4: Verify that all forms are equivalent by expanding Form B: (x - 7)(x - 13) = x^2 - 13x - 7x + 91 = x^2 - 20x + 91, which matches Form C. Expanding Form A: (x - 10)^2 - 9 = x^2 - 20x + 100 - 9 = x^2 - 20x + 91, also matches. The y-intercept is 91.

2. A drone is flying along a path modeled by the quadratic function h(t) = -2t² + 12t + 5, where h represents the drone's height in meters above ground level and t represents time in seconds. The drone operator wants to know the maximum height the drone reaches during its flight. What is the maximum height in meters? Answer: 23 Solution: Identify the quadratic function: h(t) = -2t² + 12t + 5 Since the coefficient of t² is negative (-2), the parabola opens downward, meaning the vertex represents the maximum height Find the time at which maximum height occurs using t = -b/(2a), where a = -2 and b = 12 t = -12/(2×(-2)) = -12/(-4) =…
   Full step-by-step solution

   Step 1: Identify the quadratic function: h(t) = -2t² + 12t + 5 Step 2: Since the coefficient of t² is negative (-2), the parabola opens downward, meaning the vertex represents the maximum height Step 3: Find the time at which maximum height occurs using t = -b/(2a), where a = -2 and b = 12 Step 4: t = -12/(2×(-2)) = -12/(-4) = 3 seconds Step 5: Substitute t = 3 into the original function to find the maximum height Step 6: h(3) = -2(3)² + 12(3) + 5 = -2(9) + 36 + 5 = -18 + 36 + 5 = 23 Step 7: The maximum height is 23 meters

3. Mere graphs two equivalent quadratic functions on a coordinate plane. The first function is written in vertex form: f(x) = (x - 12)^2 + 15. The second function is written in standard form: g(x) = x^2 - 24x + 159. Which form reveals the coordinates of the vertex directly, and what is the vertex? Explain which form is more useful for identifying the axis of symmetry. Answer: Vertex: (12, 15). The vertex form f(x) = (x - 12)^2 + 15 reveals the vertex and axis of symmetry directly; the axis of symmetry is x = 12. Solution: Identify the vertex form. f(x) = (x - 12)^2 + 15 is in vertex form y = (x - h)^2 + k, where (h, k) is the vertex. Here h = 12 and k = 15, so the vertex is (12, 15).
   Full step-by-step solution

   Step 1: Identify the vertex form. f(x) = (x - 12)^2 + 15 is in vertex form y = (x - h)^2 + k, where (h, k) is the vertex. Here h = 12 and k = 15, so the vertex is (12, 15). The axis of symmetry is x = h = 12. Step 2: Examine the standard form. g(x) = x^2 - 24x + 159 is in standard form y = ax^2 + bx + c. To find the vertex from standard form, you would need to complete the square or use the formula x = -b/(2a). Here a = 1, b = -24, so x = -(-24)/(2*1) = 24/2 = 12. Then y = (12)^2 - 24(12) + 159 = 144 - 288 + 159 = 15. So the vertex is also (12, 15), but it requires calculation. Step 3: Compare which form reveals properties directly. The vertex form f(x) = (x - 12)^2 + 15 immediately shows the vertex (12, 15) and the axis of symmetry x = 12 without any work. The standard form g(x) = x^2 - 24x + 159 hides these features and requires additional steps to extract them. Conclusion: The vertex form is more useful for identifying the vertex and axis of symmetry because it presents them explicitly. The answer is vertex (12, 15) and axis of symmetry x = 12.

4. A research scientist is studying the decay of a radioactive isotope used in medical imaging. The remaining mass M(t) in grams after t days is modeled by the exponential function M(t) = 80 × (1/2)^(t/15). The hospital needs to dispose of the material when only 10 grams remain. How many days will it take for the isotope to decay to this disposal level? Answer: 45 Solution: Set up the equation using the given model: 80 × (1/2)^(t/15) = 10 Divide both sides by 80 to isolate the exponential term: (1/2)^(t/15) = 10/80 = 1/8 Recognize that 1/8 can be written as a power of 1/2: 1/8 = (1/2)^3 Since the bases are the same, set the exponents equal to each other: t/15 = 3…
   Full step-by-step solution

   Step 1: Set up the equation using the given model: 80 × (1/2)^(t/15) = 10 Step 2: Divide both sides by 80 to isolate the exponential term: (1/2)^(t/15) = 10/80 = 1/8 Step 3: Recognize that 1/8 can be written as a power of 1/2: 1/8 = (1/2)^3 Step 4: Since the bases are the same, set the exponents equal to each other: t/15 = 3 Step 5: Solve for t by multiplying both sides by 15: t = 3 × 15 = 45 Step 6: The isotope will reach 10 grams after 45 days.

5. (3x² - 5x + 2) + (2x² + 4x - 7) = ? Answer: 5x² - x - 5 Solution: (3x² - 5x + 2) + (2x² + 4x - 7) Since we are adding, the parentheses don't change the signs: 3x² - 5x + 2 + 2x² + 4x - 7 - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 2 - 7 3x² + 2x² = (3 + 2)x² = 5x² -5x + 4x = (-5 + 4)x = -1x, which is -x 2 - 7 = -5 5x² - x - 5 Final answer: 5x²…
   Full step-by-step solution

   Let's add the two polynomials step by step. We have: (3x² - 5x + 2) + (2x² + 4x - 7) --- **Step 1: Remove parentheses** Since we are adding, the parentheses don't change the signs: 3x² - 5x + 2 + 2x² + 4x - 7 --- **Step 2: Group like terms** - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 2 - 7 --- **Step 3: Combine x² terms** 3x² + 2x² = (3 + 2)x² = 5x² --- **Step 4: Combine x terms** -5x + 4x = (-5 + 4)x = -1x, which is -x --- **Step 5: Combine constants** 2 - 7 = -5 --- **Step 6: Write the final polynomial** 5x² - x - 5 --- **Final answer:** 5x² - x - 5

6. Tane is a structural engineer analyzing the trajectory of a suspension bridge cable. The cable's height above the deck, in meters, is modeled by the quadratic function h(x) = -0.04x² + 1.6x + 12, where x is the horizontal distance in meters from the left tower. Tane needs to rewrite this function in vertex form, h(x) = a(x - h)² + k, to easily identify the maximum height of the cable and where it occurs. What is the vertex form of the function, and what does it reveal about the cable's maximum height and its horizontal location? Answer: h(x) = -0.04(x - 20)² + 28; maximum height 28 m at x = 20 m Solution: Start with the given function: h(x) = -0.04x² + 1.6x + 12. Factor out -0.04 from the first two terms: h(x) = -0.04(x² - 40x) + 12. Complete the square inside the parentheses.
   Full step-by-step solution

   Step 1: Start with the given function: h(x) = -0.04x² + 1.6x + 12. Step 2: Factor out -0.04 from the first two terms: h(x) = -0.04(x² - 40x) + 12. Step 3: Complete the square inside the parentheses. Take half of -40, which is -20, and square it: (-20)² = 400. Step 4: Add and subtract 400 inside the parentheses: h(x) = -0.04(x² - 40x + 400 - 400) + 12. Step 5: Rewrite as a perfect square: h(x) = -0.04[(x - 20)² - 400] + 12. Step 6: Distribute -0.04: h(x) = -0.04(x - 20)² + 16 + 12. Step 7: Combine constants: h(x) = -0.04(x - 20)² + 28. Step 8: The vertex form is h(x) = -0.04(x - 20)² + 28. This reveals that the vertex is at (20, 28), meaning the cable reaches its maximum height of 28 meters at a horizontal distance of 20 meters from the left tower. The answer is: h(x) = -0.04(x - 20)² + 28; maximum height 28 m at x = 20 m.

7. Which form of the quadratic function y = x² - 11x + 26 reveals the minimum value? Rewrite it in that form. Answer: y = (x - 5.5)² - 4.25 Solution: Start with y = x² - 11x + 26. To complete the square, take half of the coefficient of x: -11/2 = -5.5. Square it: (-5.5)² = 30.25.
   Full step-by-step solution

   Step 1: Start with y = x² - 11x + 26. Step 2: To complete the square, take half of the coefficient of x: -11/2 = -5.5. Square it: (-5.5)² = 30.25. Step 3: Add and subtract 30.25 inside the expression: y = (x² - 11x + 30.25) + 26 - 30.25. Step 4: Factor the perfect square trinomial: y = (x - 5.5)² + 26 - 30.25. Step 5: Simplify the constant: 26 - 30.25 = -4.25. Step 6: The vertex form is y = (x - 5.5)² - 4.25. This reveals the vertex at (5.5, -4.25), so the minimum value is -4.25. The answer is y = (x - 5.5)² - 4.25.