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Converting from one form to another Vertex or Root form to Standard - just FOIL and multiply: Example: y = ½(x – 2)(x + 4) y = ½(x + 2x – 8) y = ½x + x – 4 (Standard form) Standard to Vertex form – You can (a) complete the square or (b) find the axis of symmetry (x value of the vertex) and plug x back into the equation to find the y value of the vertex. (a) Completing the square: 2 Example: y = 3x – 12x – 9 a = 3, b = -12, c = -9 (y-intercept) 2 Step 1: Substitute 0 for y 0 = 3x – 12x – 9 Step 2: Move the constant 9 = 3x – 12x Step 3: Factor-out the leading coefficient 9 + ___ = 3(x – 4x + ___ ) Step 4: Add (b/2)2 to both sides Step 5: Factor the perfect square Step 6: Move the constant 9 + _12_ = 3(x – 4x + _4_ ) Note: we added 3 x 4 21 = 3 (x – 2)2 y = 3 (x – 2)2 – 21 (Vertex form) 2 2 2 (b) Find the axis of symmetry: 2 Example: y = 3x – 12x – 9 a = 3, b = -12, c = -9 (y-intercept) To find the vertex, use x = -b/2a: Here, x = -(-12)/2(3) = 2, so x = 2 is the Axis of Symmetry and the x-value of our vertex. Now, we can replace x with 2 in our equation to find the y-value of our vertex: 2 y = 3(2) – 12(2) – 9 = 12 – 24 – 9 = - 21 Now, we know the value of “a” (3) and our vertex (2,-21), so we can put this in Vertex form: 2 y = 3(x – 2) – 21 Pretty cool, huh? Vertex to Roots form: When we cannot factor the Standard form to get to the Roots form, we can set y = 0 and use the Vertex form to solve for our Roots (where the parabola crosses the x-axis): 2 Example: 2 y = 3(x – 2) – 21 0 = 3(x – 2) – 21 2 21 = 3(x – 2) Divide both sides by 3 2 7 = (x – 2) Take the square root of both sides ±√7 = x – 2 Add 2 to both sides and you have the Roots! 2±√7 = x So y = 3(x – 2+ √7)(x – 2 – √7) (Root form) Remember: Roots can be imaginary! 2 Example: y = 2(x + 1) + 2 2 0 = 2(x + 1) + 2 2 -2 = 2(x + 1) Divide both sides by 2 2 -1 = (x + 1) Take the square root of both sides Since we cannot take the square root of a negative number, our answer is the imaginary root of -1±i