For the function f(x)=-2x^2-6x-7 find:
a) an equivalent equation in the form f(x)=a(x-h)^2+k
b)the maximum or minimum (also decide which)
please help and thank you
f(x) = -2x^2 - 6x - 7
= -2(x^2 + 3x) - 7
= -2(x^2 + 3x + 9/4) - 7 + 2(9/4)
= -2(x - 3/2)^2 - 5/2
I'm sure you can answer the questions from that.
why 9/4???
nevermind I figure it out and thank you for your help
To find an equivalent equation in the form f(x) = a(x - h)^2 + k for the given function f(x) = -2x^2 - 6x - 7, we need to complete the square.
a) Equivalent Equation in Vertex Form: f(x) = a(x - h)^2 + k
1. Factor out the coefficient of x^2:
f(x) = -2(x^2 + 3x) - 7
2. To complete the square, take half of the coefficient of x (which is 3 in this case) and square it. Add and subtract this value inside the parentheses:
f(x) = -2(x^2 + 3x + (3/2)^2 - (3/2)^2) - 7
3. Rearrange the expression inside the parentheses:
f(x) = -2(x^2 + 3x + (9/4) - (9/4)) - 7
4. Simplify the expression inside the parentheses by rearranging it as a perfect square trinomial:
f(x) = -2((x + 3/2)^2 - 9/4) - 7
5. Distribute -2 across the perfect square trinomial:
f(x) = -2(x + 3/2)^2 + 9/2 - 7
6. Simplify the constant terms:
f(x) = -2(x + 3/2)^2 + 9/2 - 14/2
7. Combine the constant terms:
f(x) = -2(x + 3/2)^2 - 5/2
Therefore, an equivalent equation in the form f(x) = a(x - h)^2 + k is f(x) = -2(x + 3/2)^2 - 5/2.
b) Maximum or Minimum:
In the equation f(x) = -2(x + 3/2)^2 - 5/2, since the coefficient of the squared term (a) is negative (-2), the parabola opens downward. This means the function has a maximum point.
To find the coordinates of the maximum point, consider the vertex form of a parabola: f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
In this case, the coordinates of the vertex are (-3/2, -5/2), which represents the maximum point of the function f(x).
Therefore, for the given function f(x) = -2x^2 - 6x - 7, the equivalent equation in vertex form is f(x) = -2(x + 3/2)^2 - 5/2, and it has a maximum point at (-3/2, -5/2).