Convert the following quadratic functions to vertex form 𝑦 =2/3 𝑥^2 − 6𝑥 + 4
To convert the quadratic function 𝑦 = (2/3)𝑥^2 − 6𝑥 + 4 into vertex form, we can complete the square. The vertex form of a quadratic function is given by 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘, where (ℎ, 𝑘) represents the vertex of the parabola.
First, we can factor out the common factor (2/3) from the equation:
𝑦 = (2/3)(𝑥^2 − 9𝑥) + 4
Next, we want to complete the square for the expression inside the parentheses. To do this, we need to take half of the coefficient of 𝑥, square it, and add it inside the parentheses:
𝑦 = (2/3)(𝑥^2 − 9𝑥 + (9/2)^2) + 4 − (2/3)(9/2)^2
Simplifying further:
𝑦 = (2/3)(𝑥^2 − 9𝑥 + 81/4) + 4 − (2/3)(81/4)
𝑦 = (2/3)(𝑥^2 − 9𝑥 + 81/4) + 4 − 27/2
𝑦 = (2/3)(𝑥^2 − 9𝑥 + 81/4) + 8/2 − 27/2
𝑦 = (2/3)(𝑥^2 − 9𝑥 + 81/4) + (8 − 27)/2
𝑦 = (2/3)(𝑥^2 − 9𝑥 + 81/4) + (-19/2)
We can rewrite the expression inside the parentheses as a perfect square trinomial:
𝑦 = (2/3)(𝑥 - 9/2)^2 − 19/2
So, the quadratic function 𝑦 = (2/3)𝑥^2 − 6𝑥 + 4 in vertex form is 𝑦 = (2/3)(𝑥 - 9/2)^2 − 19/2.