The roots of a quadratic equation are -1 and -5, and a=1.
A.) What are the factors of the quadratic expression?
B.) Write the function in standard form using function notation.
C.) What is the degree of this function?
D.) Identify the vertex
y = (x+1)(x+5)
To find the factors of a quadratic equation, you need to find the quadratic expression in factored form.
A.) The factored form of a quadratic equation with roots -1 and -5 can be written as (x + 1)(x + 5). These are the factors of the quadratic expression.
B.) Writing the function in standard form using function notation requires multiplying out the factors.
The function in standard form would be: f(x) = (x + 1)(x + 5)
C.) The degree of a quadratic function is 2, because it is a second-degree polynomial. In this case, the degree of the function is 2.
D.) To identify the vertex of a quadratic function, you need to first convert it into vertex form, which is f(x) = a(x - h)^2 + k.
Since a = 1, the vertex form is f(x) = (x - h)^2 + k.
The vertex of a quadratic function with roots -1 and -5 lies on the axis of symmetry, which is the average of the two roots.
The axis of symmetry is ( -1 + (-5) ) / 2 = -6 / 2 = -3.
So the x-coordinate of the vertex is -3.
To find the y-coordinate, substitute this x-coordinate into the quadratic function:
f(-3) = (-3 + 1)(-3 + 5) = (-2)(2) = -4.
Therefore, the vertex is (-3, -4).