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Equivalent Forms

Grade 9 · Algebra · Worksheet 2

  1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The hypotenuse of this triangle has length 5√5 units. What is the value of x? Answer: ______________
  2. Isabella is designing a parabolic arch for a new entrance to a botanical garden. The shape of the arch can be modeled by the quadratic function h(x) = -x² + 7x - 12, where h(x) is the height in meters at a horizontal distance x meters from the left base. The construction team needs to know the horizontal distances where the arch meets the ground (the zeros of the function). Rewrite h(x) in its factored form, and explain which form — the original standard form or the factored form — more clearly reveals the zeros of the function. Then, state the two x-values where the arch touches the ground. Answer: ______________
  3. 2x² - 8x + 6 = ? when x = 5 Answer: ______________
  4. Which form of the quadratic expression 4x² + 20x + 25 reveals its minimum value, and what is that minimum value? Answer: ______________
  5. √(x² - 6x + 9) + |x - 3| = ? when x = 7 Answer: ______________
  6. Emma draws a parabola on a coordinate plane. She knows that the parabola's vertex is at (3, -1) and that it passes through the point (1, 7). She wants to express the equation of the parabola in two equivalent forms: vertex form and standard form. Which form immediately reveals the coordinates of the vertex, and what is the equation of the parabola in that form? Answer: ______________
  7. A research scientist is studying the decay of a radioactive isotope. The amount remaining after t years is modeled by the exponential function A(t) = 800 × (1/2)^(t/12). The scientist needs to determine when only 100 grams of the isotope will remain. How many years will it take for the isotope to decay to this amount? Answer: ______________
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Answer Key & Explanations

Equivalent Forms · Grade 9 · Worksheet 2

  1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The hypotenuse of this triangle has length 5√5 units. What is the value of x? Answer: 5 Solution: Identify the coordinates and side lengths. The vertices are (0,0), (x,0), and (0,2x). From (0,0) to (x,0): length = x.
    Full step-by-step solution

    Step 1: Identify the coordinates and side lengths. The vertices are (0,0), (x,0), and (0,2x). From (0,0) to (x,0): length = x. From (0,0) to (0,2x): length = 2x. These are the two legs of the right triangle, with the right angle at (0,0). Step 2: Apply the Pythagorean theorem. The hypotenuse length is given as 5√5. So: (leg1)^2 + (leg2)^2 = (hypotenuse)^2 x^2 + (2x)^2 = (5√5)^2 Step 3: Simplify the equation. x^2 + 4x^2 = 25 * 5 5x^2 = 125 Step 4: Solve for x^2. x^2 = 125 / 5 x^2 = 25 Step 5: Solve for x. x = √25 x = 5 (since x > 0 in this geometric context) Final answer: x = 5

  2. Isabella is designing a parabolic arch for a new entrance to a botanical garden. The shape of the arch can be modeled by the quadratic function h(x) = -x² + 7x - 12, where h(x) is the height in meters at a horizontal distance x meters from the left base. The construction team needs to know the horizontal distances where the arch meets the ground (the zeros of the function). Rewrite h(x) in its factored form, and explain which form — the original standard form or the factored form — more clearly reveals the zeros of the function. Then, state the two x-values where the arch touches the ground. Answer: 3 and 4 Solution: The function is h(x) = -x² + 7x - 12. Factor out -1 to make factoring easier: h(x) = -(x² - 7x + 12). Factor the quadratic inside: x² - 7x + 12.
    Full step-by-step solution

    Step 1: The function is h(x) = -x² + 7x - 12. Factor out -1 to make factoring easier: h(x) = -(x² - 7x + 12). Step 2: Factor the quadratic inside: x² - 7x + 12. Find two numbers that multiply to 12 and add to -7. These numbers are -3 and -4 because (-3) × (-4) = 12 and (-3) + (-4) = -7. Step 3: So x² - 7x + 12 = (x - 3)(x - 4). Then h(x) = -(x - 3)(x - 4). Step 4: The factored form h(x) = -(x - 3)(x - 4) clearly reveals the zeros because when either factor equals zero, the entire product is zero. Set x - 3 = 0 → x = 3, and x - 4 = 0 → x = 4. Step 5: The standard form h(x) = -x² + 7x - 12 does not directly show the zeros; you must factor or use the quadratic formula to find them. The factored form directly reveals the zeros. Step 6: Therefore, the arch touches the ground at x = 3 meters and x = 4 meters from the left base.

  3. 2x² - 8x + 6 = ? when x = 5 Answer: 16 Solution: Substitute x = 5 into the expression: 2(5)² - 8(5) + 6 Calculate the exponent first: 2(25) - 8(5) + 6 Perform multiplication: 50 - 40 + 6 Perform addition and subtraction from left to right: 50 - 40 = 10, then 10 + 6 = 16 The answer is 16.
    Full step-by-step solution

    Step 1: Substitute x = 5 into the expression: 2(5)² - 8(5) + 6 Step 2: Calculate the exponent first: 2(25) - 8(5) + 6 Step 3: Perform multiplication: 50 - 40 + 6 Step 4: Perform addition and subtraction from left to right: 50 - 40 = 10, then 10 + 6 = 16 The answer is 16.

  4. Which form of the quadratic expression 4x² + 20x + 25 reveals its minimum value, and what is that minimum value? Answer: 0 Solution: Recognize that 4x² + 20x + 25 can be written as (2x)² + 2(2x)(5) + 5², which is a perfect square trinomial. Factor it as (2x + 5)².
    Full step-by-step solution

    Step 1: Recognize that 4x² + 20x + 25 can be written as (2x)² + 2(2x)(5) + 5², which is a perfect square trinomial. Step 2: Factor it as (2x + 5)². Step 3: The expression (2x + 5)² is always greater than or equal to 0, with its minimum value occurring when 2x + 5 = 0. Step 4: Solve 2x + 5 = 0 → 2x = -5 → x = -5/2. Step 5: Substitute x = -5/2 into (2x + 5)²: (2(-5/2) + 5)² = (-5 + 5)² = 0² = 0. The minimum value is 0.

  5. √(x² - 6x + 9) + |x - 3| = ? when x = 7 Answer: 8 Solution: Substitute x = 7 into the expression: √(7² - 6×7 + 9) + |7 - 3| Calculate inside the square root: 7² = 49, 6×7 = 42, so 49 - 42 + 9 = 16 Simplify the square root: √16 = 4 Calculate the absolute value: |7 - 3| = |4| = 4 Add the results: 4 + 4 = 8 The answer is 8.
    Full step-by-step solution

    Step 1: Substitute x = 7 into the expression: √(7² - 6×7 + 9) + |7 - 3| Step 2: Calculate inside the square root: 7² = 49, 6×7 = 42, so 49 - 42 + 9 = 16 Step 3: Simplify the square root: √16 = 4 Step 4: Calculate the absolute value: |7 - 3| = |4| = 4 Step 5: Add the results: 4 + 4 = 8 The answer is 8.

  6. Emma draws a parabola on a coordinate plane. She knows that the parabola's vertex is at (3, -1) and that it passes through the point (1, 7). She wants to express the equation of the parabola in two equivalent forms: vertex form and standard form. Which form immediately reveals the coordinates of the vertex, and what is the equation of the parabola in that form? Answer: y = 2(x - 3)^2 - 1 Solution: The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. Given vertex (3, -1), we have h = 3 and k = -1, so the equation is y = a(x - 3)^2 - 1. The parabola passes through (1, 7).
    Full step-by-step solution

    Step 1: The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. Given vertex (3, -1), we have h = 3 and k = -1, so the equation is y = a(x - 3)^2 - 1. Step 2: The parabola passes through (1, 7). Substitute x = 1 and y = 7 into the equation: 7 = a(1 - 3)^2 - 1 7 = a(-2)^2 - 1 7 = a(4) - 1 7 = 4a - 1 Step 3: Solve for a: 7 + 1 = 4a 8 = 4a a = 8/4 a = 2 Step 4: So the vertex form is y = 2(x - 3)^2 - 1. This form immediately reveals the vertex (3, -1). The answer is y = 2(x - 3)^2 - 1.

  7. A research scientist is studying the decay of a radioactive isotope. The amount remaining after t years is modeled by the exponential function A(t) = 800 × (1/2)^(t/12). The scientist needs to determine when only 100 grams of the isotope will remain. How many years will it take for the isotope to decay to this amount? Answer: 36 Solution: Step 1: Set up the equation using the given function: 800 × (1/2)^(t/12) = 100 Step 2: Divide both sides by 800: (1/2)^(t/12) = 100/800 = 1/8 Step 3: Recognize that 1/8 = (1/2)^3 Step 4: Set the exponents equal: t/12 = 3 Step 5: Multiply both sides by 12: t = 3 × 12 = 36 Step 6: Verify by…
    Full step-by-step solution

    Step 1: Set up the equation using the given function: 800 × (1/2)^(t/12) = 100 Step 2: Divide both sides by 800: (1/2)^(t/12) = 100/800 = 1/8 Step 3: Recognize that 1/8 = (1/2)^3 Step 4: Set the exponents equal: t/12 = 3 Step 5: Multiply both sides by 12: t = 3 × 12 = 36 Step 6: Verify by substituting back: 800 × (1/2)^(36/12) = 800 × (1/2)^3 = 800 × 1/8 = 100 The answer is 36 years.