quadratic functions to vertex form by completing the square.𝑦 =2/3𝑥^2 − 6𝑥 + 4
To convert the quadratic function 𝑦 = (2/3)𝑥^2 − 6𝑥 + 4 to vertex form by completing the square, follow these steps:
Step 1: Factor out the leading coefficient (2/3) from the terms involving 𝑥^2 and 𝑥.
𝑦 = (2/3)(𝑥^2 − 9𝑥) + 4
Step 2: Take half of the coefficient of 𝑥 and square it. Add and subtract this value inside the parentheses.
𝑦 = (2/3)(𝑥^2 − 9𝑥 + (−9/2)^2 − (−9/2)^2) + 4
Step 3: Simplify inside the parentheses.
𝑦 = (2/3)(𝑥^2 − 9𝑥 + 81/4 − 81/4) + 4
Step 4: Simplify the terms inside the parentheses.
𝑦 = (2/3)((𝑥 − 9/2)^2 − 81/4) + 4
Step 5: Distribute the leading coefficient back in.
𝑦 = (2/3)(𝑥 − 9/2)^2 − (2/3)(81/4) + 4
Step 6: Simplify the constant terms.
𝑦 = (2/3)(𝑥 − 9/2)^2 − 54/2 + 4
Step 7: Further simplify the constant terms.
𝑦 = (2/3)(𝑥 − 9/2)^2 − 27 + 4
Step 8: Combine like terms.
𝑦 = (2/3)(𝑥 − 9/2)^2 − 23
The quadratic function 𝑦 = (2/3)𝑥^2 − 6𝑥 + 4 in vertex form after completing the square is 𝑦 = (2/3)(𝑥 − 9/2)^2 − 23. The vertex is (9/2, -23).