GPS Algebra Converting from Standard to Vertex Form

The standard form of a quadratic equation is ax2  bx  c  0 . The vertex form of a quadratic equation is a(x  h)2  k  0 where (h, k) is your vertex.

Find the vertices of the following quadratic equations: 1. (x  5) 2  7  0 2. 2(x  4) 2  8  0 3.  (x  6) 2  6  0

Once a quadratic equation is in vertex form it becomes easier to draw an accurate parabola. But how do you convert a quadratic equation from standard form to vertex form? Follow these steps:

STEP 1: Make sure that one side of the equation is equal to 0. STEP 2: Find the a value. The a value in vertex form is the same as the a value in standard form. STEP 3: Find the h value. In order to find the h value of a quadratic equation use the b formula  , where a and b are the values from the standard form. 2a STEP 4: Find the k value. In order to find the k value of a quadratic equation, substitute your h value into your quadratic equation and solve. STEP 5: Place all of your values into the vertex form of a quadratic equation.

Ex 1: x 2  2x  4 STEP 1: Add 4 to both sides of the equation to make it x 2  2x  4  0 STEP 2: In the standard form, a = 1. Therefore, in vertex form a = 1 as well. b (2) 2 STEP 3: h =  when b is -2, and a is 1. Therefore h =    1. 2a 2(1) 2 STEP 4: To find the value of k, substitute 1 into the equation x 2  2x  4 . (1) 2  2(1)  4 , 1 2  4 = 3. STEP 5: The vertex form of the standard quadratic equation x 2  2x  4  0 is (x 1) 2  3  0 .

Ex 2: 4x 2 16x  2  0 STEP 1: 4x 2 16x  2  0 STEP 2: a = 4 b 16 16 STEP 3: h =  =  =   2 2a 2(4) 8 STEP 4: 4(2)2 16(2)  2 = 4(4) 16(2)  2 = 16  32  2 = – 14 STEP 5: 4x 2 16x  2  0 is 4(x  2) 2 14  0 in vertex form.

Now you try a few: 1. x 2  2x  2  0 2. 2x 2 12x  25  0 3. x 2 12x  32  0

4. x 2  4x  7  0 5. x 2  6x 16  0 6. x 2  3x  3  0 Vertex to standard form

To change a quadratic equation from vertex to standard form we must simply follow order of operations. PEMDAS Step 1: Foil the squared binomial. Step 2: Distribute the “a” to the answer of your squared binomial. Step 3: Add the “k” to the other constant.

Change these equations from vertex to standard form. 7. y = 8(x + 1)2 – 7 8. Y = -(x + 3)2 + 6

9. y = -2(x – 1)2 – 4 10. Y = (x – 5)2 + 19