how can I figure out the formula. example. I need to transform the following equation y = x2 + 4 - 8 into vertex form and then I need to state the coordinates of the vertex of the parabola. I forgot how to arrive at the answer
I will assume there is a typo and you meant
y = x^2 + 4x - 8
take 1/2 the coefficient of the x term, square it, then add and then subtract it
( 1/2 of 4 is 2 and then squared would be 4)
y = x^2 + 4x + 4 - 4 - 8
the first three terms are now a perfect square
y = (x+2)^2 - 12
compare this with
y = a(x-p)^2 + q which has vertex (p,q) and you can conclude that the vertex of
y = (x+2)^2 - 12 is (-2,-12)
To transform the given equation into vertex form, also known as completing the square, follow these steps:
Step 1: Write down the equation in the standard form: y = ax^2 + bx + c. In this case, the equation given is y = x^2 + 4x - 8.
Step 2: Group the x-terms together: y = (x^2 + 4x) - 8.
Step 3: Focus on the terms inside the parentheses (x^2 + 4x) and complete the square.
a. Take half of the coefficient of the x-term (4), which is 2, and square it: (2)^2 = 4.
b. Add this value inside the parentheses, and subtract it outside the parentheses to maintain the equation's balance: y = (x^2 + 4x + 4) - 4 - 8.
c. Simplify the equation: y = (x^2 + 4x + 4) - 12.
Step 4: Factor the polynomial inside the parentheses by finding perfect square trinomials:
a. The binomial (x^2 + 4x + 4) can be factored as (x + 2)^2.
b. Rewrite the equation: y = (x + 2)^2 - 12.
Step 5: Simplify the equation further if needed. In this case, the equation is already in vertex form.
Now, to find the coordinates of the vertex, you can compare the equation to the vertex form: y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
In this case, the equation is y = (x + 2)^2 - 12. Comparing it to the vertex form, you can see that h = -2 and k = -12. Therefore, the coordinates of the vertex are (-2, -12).