Questions: Consider the following quadratic function. f(x)=-3(x+1)^2+5 Identify the vertex and the axis of symmetry. vertex (x, y)= axis of symmetry Determine whether the vertex yields a relative and absolute maximum or minimum. relative and absolute maximum relative and absolute minimum

Solution Steps

To find the vertex of the quadratic function in vertex form \( f(x) = a(x-h)^2 + k \), identify \( h \) and \( k \) from the equation. The vertex is \((h, k)\). The axis of symmetry is the vertical line \( x = h \). Since the coefficient \( a \) is negative, the parabola opens downwards, indicating that the vertex is a point of relative and absolute maximum.

The vertex of the quadratic function \( f(x) = -3(x+1)^{2} + 5 \) is given by the coordinates \( (h, k) \). From the function, we have: \[ h = -1, \quad k = 5 \] Thus, the vertex is: \[ \text{Vertex} = (-1, 5) \]

The axis of symmetry for a quadratic function in vertex form is given by the line \( x = h \). Therefore, the axis of symmetry is: \[ \text{Axis of Symmetry} = x = -1 \]

Since the coefficient \( a = -3 \) is negative, the parabola opens downwards. This indicates that the vertex represents both a relative and absolute maximum point of the function.

Final Answer