Select the correct vertex form of the following equation. show all your work
y=x2+6x−9
answer : y=(x−3)2−4
To convert the given equation y = x^2 + 6x - 9 into vertex form, we need to complete the square.
We can do this by taking the coefficient of x, which is 6, dividing it by 2 (giving us 3), and then squaring it (giving us 9). We then add and subtract this value within the equation.
y = x^2 + 6x - 9
= (x^2 + 6x + 9) - 9 - 9
= (x^2 + 6x + 9) - 18
= (x + 3)^2 - 18
Therefore, the correct vertex form of the equation is y = (x + 3)^2 - 18.
To convert the equation y = x^2 + 6x - 9 into vertex form, follow these steps:
Step 1: Complete the square for the x terms:
To complete the square, take half of the coefficient of the x term (6) and square it.
Half of 6 is 3, and 3 squared is 9.
So, add 9 to both sides of the equation:
y + 9 = x^2 + 6x + 9 - 9
y + 9 = x^2 + 6x
Step 2: Rewrite the equation as a perfect square trinomial:
Now, rewrite the equation as a perfect square trinomial by adding the square of half the coefficient of x to both sides of the equation.
y + 9 + 9 = x^2 + 6x + 9
y + 18 = (x + 3)^2
Step 3: Rearrange the equation in vertex form:
To get the equation in vertex form, isolate y by subtracting 18 from both sides of the equation:
y + 18 - 18 = (x + 3)^2 - 18
y = (x + 3)^2 - 18
So, the correct vertex form of the equation y = x^2 + 6x - 9 is y = (x - 3)^2 - 4.
To find the vertex form of the given equation, you need to complete the square. Here are the steps to do it:
Step 1: Group the terms involving the variable 'x':
y = x^2 + 6x - 9
Step 2: Take half of the coefficient of 'x' and square it:
(6/2)^2 = 9
Step 3: Add the result from step 2 to both sides of the equation:
y + 9 = x^2 + 6x + 9
Step 4: Rewrite the right side of the equation as a perfect square trinomial:
y + 9 = (x^2 + 6x + 9)
Step 5: Factor the trinomial in parentheses and rewrite the equation:
y + 9 = (x + 3)^2
Step 6: To isolate 'y', subtract 9 from both sides of the equation:
y = (x + 3)^2 - 9
Simplifying, we get:
y = (x + 3)^2 - 9
So, the correct vertex form of the equation y = x^2 + 6x - 9 is y = (x - 3)^2 - 4.
The vertex form represents a parabola in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Comparing this with our equation, we can see that the vertex is (-3, -4).