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 features | most useful form of f(x) --- | --- the vertex of the function | the x - intercepts of the function | the y - intercept | the axis of symmetry | f(x)=x² - 6x - 16, f(x)=(x - 3)² - 25, f(x)=(x - 8)(x + 2)

Explanation:

Step1: Analyze each key feature and form

• The person of the function (vertex): The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. So \( f(x)=(x - 4)^2 - 24 \) (vertex form) is most useful.
• The x - intercepts (roots): The factored form \( f(x) = a(x - r_1)(x - r_2) \) shows the roots directly. So \( f(x)=(x - 8)(x + 2) \) (factored form) is most useful.
• The y - intercept: The standard form \( f(x)=ax^2+bx + c \) has \( c \) as the y - intercept. So \( f(x)=x^2 - 6x - 16 \) (standard form) is most useful.
• The axis of symmetry: The vertex form \( f(x) = a(x - h)^2 + k \) has the axis of symmetry \( x = h \). So \( f(x)=(x - 4)^2 - 24 \) (vertex form) is most useful.

Step2: Match each key feature with the form

• For "The person of the function (vertex)": \( f(x)=(x - 4)^2 - 24 \)
• For "The x - intercepts of the function": \( f(x)=(x - 8)(x + 2) \)
• For "The y - intercept": \( f(x)=x^2 - 6x - 16 \)
• For "The axis of symmetry": \( f(x)=(x - 4)^2 - 24 \)

Answer:

Key Feature	Most Useful Form of \( f(x) \)
The x - intercepts	\( f(x)=(x - 8)(x + 2) \)
The y - intercept	\( f(x)=x^2 - 6x - 16 \)
The axis of symmetry	\( f(x)=(x - 4)^2 - 24 \)