The quadratic function [tex]f(x)=a(x-h)^2+k, a \neq 0[/tex], is in $\square$ form. The graph of [tex]f[/tex] is called a $\square$ whose vertex is the point $\square$. The graph opens upward if $\square$ and opens downward if $\square$.
Answer (2)
The quadratic function f ( x ) = a ( x − h ) 2 + k is in vertex form. The graph of this function is called a parabola with its vertex at the point ( h , k ) . It opens upward if 0"> a > 0 and downward if a < 0 .
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The quadratic function is in vertex form.
The graph of the function is a parabola.
The vertex of the parabola is at point ( h , k ) .
The parabola opens upward if 0"> a > 0 and downward if a < 0 .
The quadratic function f ( x ) = a ( x − h ) 2 + k , a = 0 , is in vertex form. The graph of f is called a parabola whose vertex is the point ( h , k ) . The graph opens upward if 0}"> a > 0 and opens downward if a < 0 .
Explanation
Understanding the Quadratic Function The given quadratic function is in the form f ( x ) = a ( x − h ) 2 + k , where a = 0 . We need to identify the name of this form, the shape of the graph, the vertex of the graph, and the conditions for the graph to open upwards or downwards.
Identifying the Form The form f ( x ) = a ( x − h ) 2 + k is called the vertex form of a quadratic function.
Identifying the Graph The graph of a quadratic function is called a parabola.
Identifying the Vertex The vertex of the parabola is the point ( h , k ) .
Determining the Direction of Opening The parabola opens upwards if 0"> a > 0 and opens downwards if a < 0 .
Final Answer Therefore, the quadratic function f ( x ) = a ( x − h ) 2 + k , a = 0 , is in vertex form. The graph of f is called a parabola whose vertex is the point ( h , k ) . The graph opens upward if 0"> a > 0 and opens downward if a < 0 .
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to design parabolic mirrors in telescopes or antennas, optimizing the focus of light or radio waves. Architects apply this knowledge to create arches and suspension bridges, ensuring structural stability and aesthetic appeal. Moreover, economists use quadratic models to analyze cost and revenue curves, helping businesses maximize profits and minimize losses. These applications demonstrate the practical significance of understanding quadratic functions and their properties.
