write each function in vertex form f(x) =-2x^2 =6x + 10
repeat the equation again, is it
f(x) = -2x^2 = 6x + 10
which is it? why 2 equal signs?
if you mean
y = -2 x^2 - 6 x + 10
then complete the square
-2 x^2 -6 x = y - 10
x^2 + 3x = -y/2 + 5
x^2 + 3 x + 9/4 = -y/2 +5 + 9/4
(x+3/2)^2 = -(1/2) (y -29/2)
vertex at
x = -3/2 and y = 29/2
yes that's the equation
To write the given function in vertex form, also known as the standard form, we need to complete the square. The standard form of a quadratic function is typically expressed as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Let's start by rearranging the given equation to group the x-terms together and move the constant term to the other side of the equation:
f(x) = -2x^2 - 6x - 10
Now, to complete the square and convert this equation into vertex form:
Step 1: Factor out the common coefficient from the x-squared term:
f(x) = -2(x^2 + 3x) - 10
Step 2: To complete the square, we need to find the value that, when added to the expression inside the parentheses, would make it a perfect square trinomial. Taking half of the coefficient of the x-term and squaring it will give us this value. In this case, half of 3 is 3/2, and squaring it gives us 9/4.
Step 3: Add the value determined in step 2 inside the parentheses, but since we cannot modify the equation's value, we must also subtract its equivalent outside the parentheses:
f(x) = -2(x^2 + 3x + 9/4 - 9/4) - 10
Step 4: Now, let's express the perfect square trinomial within the parentheses:
f(x) = -2((x + 3/2)^2 - 9/4) - 10
Step 5: Distribute the -2 to both terms within the parentheses:
f(x) = -2(x + 3/2)^2 + 9/2 - 10
Step 6: Simplify the constant terms:
f(x) = -2(x + 3/2)^2 + 9/2 - 20/2
Step 7: Combine the constant terms further:
f(x) = -2(x + 3/2)^2 - 11/2
That's it! The function f(x) = -2x^2 + 6x + 10 can be written in vertex form as f(x) = -2(x + 3/2)^2 - 11/2. The vertex of this quadratic function is (-3/2, -11/2).