Converting Between Standard Form and Vertex Form Unit 4 - Quadratics

Two Forms of a Quadratic y = ax2 + bx + c a = # in front of x2 b = # in front of x c = # without a variable c is always the y- intercept Can be graphed by a table of values, finding the vertex, or by graphing calculator y = a(x – h)2 + k a is the # in front of x2 h is the x-value of the vertex k is the y-value of the vertex (h, k) represents the vertex Can be graphed by transformations Standard Form Vertex Form

Converting From Standard Form to Vertex Form To convert from Standard Form y = ax2 + bx + c to Vertex Form y = a(x – h)2 + k you must find the vertex using x = -b/2a x = -b/2a finds the x-value of the vertex, which is h. Then you plug the x-value you found into the equation in standard form to find the y-value of the vertex. The a comes from the # in front of the x2. Now rewrite the equation using the values of h, k, and a in vertex form.

Example 1: Convert from standard form to vertex form: y = 3x2 + 6x - 1 Use x = -b/2a a = 3, b = 6 x = -6/2(3) x = -6/6 x = -1, which means h = - 1 We know x = -1, so we plug that into the standard form y = 3(-1)2 + 6(-1) – 1 y = 3(1) + 6(-1) – 1 y = 3 – 6 – 1 y = -3 – 1 y = -4, which means k = -4 To Find the x-value (h) To Find the y-value (k)

Writing Out the Final Answer k = -4 a = 3 y = a(x – h)2 + k y = 3(x – (-1))2 – 4 y = 3(x + 1)2 – 4

You Try 1: Convert from standard form to vertex form: y = -2x2 + 12x – 12 Use x = -b/2a a = -2, b = 12 x = -12/2(-2) x = -12/-4 x = 3, which means h = 3 We know x = 3, so we plug that into the standard form y = -2(3)2 + 12(3) – 12 y = -2(9) + 12(3) – 12 y = -18 + 36 – 12 y = 18 – 12 y = 6, which means k = 6 To Find the x-value (h) To Find the y-value (k)

Writing Out the Final Answer k = 6 a = -2 y = a(x – h)2 + k y = -2(x – (3))2 + 6 y = -2(x - 3)2 + 6

Converting From Vertex Form to Standard Form To convert from Vertex Form y = a(x – h)2 + k to Standard Form y = ax2 + bx + c you must simplify the vertex form of the equation by following ORDER OF OPERATIONS. Parentheses Exponents Multiplication/Division Addition/Subtraction

Example 2: Convert from vertex form to standard form: y = 4(x – 2)2 + 3 First step is to square (x – 2) (x – 2)(x – 2) x2 – 2x – 2x + 4 x2 – 4x + 4 Second step is the distribute 4 4(x2 – 4x + 4) 4x2 – 16x + 16 Third step is to add 3 y = 4x2 – 16x + 19 . . . This is the final answer