Question

find the y - intercept, the axis of symmetry, and the vertex of the graph of the function
$f(x) = -2x^2 - 4x - 4$
the y - intercept is (0, - 4). (type an ordered pair.)
the axis of symmetry is $x = square$. (simplify your answer. type an equation.)

Explanation:

Step1: Find the y - intercept

To find the y - intercept of a function \(y = f(x)\), we set \(x = 0\) and solve for \(y\).
For the function \(f(x)=-2x^{2}-4x - 4\), when \(x = 0\), we have:
\(f(0)=-2(0)^{2}-4(0)-4=-4\)
So the y - intercept is the point \((0,-4)\).

Step2: Find the axis of symmetry

For a quadratic function in the form \(f(x)=ax^{2}+bx + c\) (\(a
eq0\)), the equation of the axis of symmetry is given by the formula \(x =-\frac{b}{2a}\).
In the function \(f(x)=-2x^{2}-4x - 4\), we have \(a=-2\) and \(b = - 4\).
Substitute \(a=-2\) and \(b=-4\) into the formula for the axis of symmetry:
\(x=-\frac{-4}{2\times(-2)}=\frac{4}{-4}=-1\)
So the axis of symmetry is \(x=-1\).

Step3: Find the vertex

The x - coordinate of the vertex is the same as the axis of symmetry, so \(x=-1\).
To find the y - coordinate of the vertex, we substitute \(x = - 1\) into the function \(f(x)\):
\(f(-1)=-2(-1)^{2}-4(-1)-4=-2 + 4-4=-2\)
So the vertex is the point \((-1,-2)\).

Answer:

• y - intercept: \((0,-4)\)
• Axis of symmetry: \(x=-1\)
• Vertex: \((-1,-2)\)