Which of the following means the same as the solutions of a quadratic function?
(Select all that apply.)
roots
zeros
x-intercepts
y-intercepts
vertex
axis of symetry
roots
zeros
x-intercepts
Which function has a vertex at (3, -9) and zeros at x = 0 and x = 6?
A quadratic function with a vertex at (3, -9) and zeros at x = 0 and x = 6 can be written in the vertex form of a quadratic function as:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex.
Given that the vertex is at (3, -9), we have h = 3 and k = -9.
Since the zeros are at x = 0 and x = 6, we can express the function in factored form as:
f(x) = a(x - 0)(x - 6) = a(x)(x - 6)
To find the value of 'a' we can substitute the vertex into the equation:
-9 = a(3)(3 - 6)
-9 = -9a
a = 1
Therefore, the quadratic function is:
f(x) = (x)(x - 6)
f(x) = x^2 - 6x
Which function is equivalent to
f\left(x\right)=-4\left(x+7\right)^{2}-6
To find the equivalent function to \(f(x) = -4(x+7)^2 - 6\), we need to expand and simplify the given function.
\(f(x) = -4(x+7)^2 - 6\)
\(f(x) = -4(x^2 + 14x + 49) - 6\)
\(f(x) = -4x^2 - 56x - 196 - 6\)
\(f(x) = -4x^2 - 56x - 202\)
Therefore, the equivalent function is \(f(x) = -4x^2 - 56x - 202\).