How do i convert quadratic functions from standard to vertex form?
y=2x^2+12x+14
y=x^2+x+1
complete the square
y=2x^2+12x+14
you need the coefficient of x^2 to be 1 so divide everything by 2
y/2 = x^2 + 6 x + 7
move the constant to the left
y/2 - 7 = x^2 + 6 x
add the square of half the coef of x to both sides, in other words 3^2 or 9
y/2 + 2 = x^2 + 6 x + 9
or
y/2 + 2 = (x+3)^2
that is a parabola symmetric about the line x = -3
when x = -3
y/2 + 2 = 0
so y = - 4
so
vertex at (-3 , -4)
y = x^2 + 1 x + 1
y-1 = x^2 + 1 x
y-1 + 1/4 = x^2 + x + 1/4
y - 3/4 = (x+1/2)^2
vertex at (-1/2 , 3/4)
To convert a quadratic function from standard form to vertex form, you can follow the steps below:
Standard form: y = ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k
Step 1: Identify the values of a, b, and c in the standard form equation.
The equation given is y = 2x^2 + 12x + 14. Here, a = 2, b = 12, and c = 14.
Step 2: Use the formula to find the x-coordinate of the vertex.
The x-coordinate of the vertex (h) can be found using the formula: h = -b / (2a). In this case, h = -12 / (2 * 2) = -12 / 4 = -3.
Step 3: Substitute the x-coordinate of the vertex into the equation to find the y-coordinate.
The y-coordinate of the vertex (k) can be found by substituting the calculated value of h into the equation: k = a(-3)^2 + b(-3) + c. Substituting the values, we get k = 2 * (-3)^2 + 12 * (-3) + 14 = 26.
Step 4: Rewrite the equation in vertex form using the values of (h, k).
Now that we have the values of h and k, we can rewrite the equation in vertex form. In this case, the vertex form equation is y = 2(x + 3)^2 + 26.
Repeat the same steps for the second quadratic equation:
Given equation: y = x^2 + x + 1
Step 1: Identify the values of a, b, and c in the standard form equation.
In this equation, a = 1, b = 1, and c = 1.
Step 2: Calculate the x-coordinate of the vertex.
Using the formula h = -b / (2a), the x-coordinate of the vertex is h = -1 / (2 * 1) = -1 / 2.
Step 3: Substitute the x-coordinate of the vertex into the equation to find the y-coordinate.
k = a(-1/2)^2 + b(-1/2) + c = 1 * (-1/2)^2 + 1 * (-1/2) + 1 = 1/4 - 1/2 + 1 = 1/4 - 2/4 + 4/4 = 3/4.
Step 4: Rewrite the equation in vertex form using the values of (h, k).
The vertex form equation, using the values (h, k), is y = (x + 1/2)^2 + 3/4.