Question

the table describes four key features of a quadratic function.
move the functions into the table to show which form of ( f(x) ) is most useful to identify each key feature.

key feature	most useful form of ( f(x) )
the coordinates of the vertex
the ( y )-intercept
the axis of symmetry

( f(x) = x^2 - 6x - 16 )
( f(x) = (x - 3)^2 - 25 )
( f(x) = (x - 8)(x + 2) )

Explanation:

Step1: Identify Vertex Form Use

The vertex form of a quadratic is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. So for the coordinates of the vertex, we use \( f(x) = (x - 3)^2 - 25 \) (since it's in vertex form, \( h = 3 \), \( k = -25 \), vertex is \((3, -25)\)).

Step2: Identify Standard Form for y - intercept

The standard form is \( f(x)=ax^{2}+bx + c \), and the y - intercept is \( c \) (when \( x = 0 \)). For \( f(x)=x^{2}-6x - 16 \), when \( x = 0 \), \( f(0)=- 16 \), so standard form is best for y - intercept.

Step3: Identify Vertex Form for Axis of Symmetry

The axis of symmetry of a quadratic in vertex form \( f(x)=a(x - h)^2 + k \) is \( x = h \). For \( f(x)=(x - 3)^2-25 \), the axis of symmetry is \( x = 3 \), so vertex form is best for axis of symmetry.

Filling the Table:

Key Feature	Most Useful Form of \( f(x) \)
the coordinates of the vertex	\( f(x)=(x - 3)^2 - 25 \) (vertex form, vertex is \( (3,-25) \))
the y - intercept	\( f(x)=x^{2}-6x - 16 \) (standard form, y - intercept is \( - 16 \))
the axis of symmetry	\( f(x)=(x - 3)^2 - 25 \) (vertex form, axis of symmetry is \( x = 3 \))

Answer:

• Coordinates of the vertex: \( f(x)=(x - 3)^2 - 25 \)
• y - intercept: \( f(x)=x^{2}-6x - 16 \)
• Axis of symmetry: \( f(x)=(x - 3)^2 - 25 \)