Ali graphs the function $f(x)=-(x+2)^2-1$ as shown.
Which best describes the error in the graph?
A. The axis of symmetry should be $x=-1$.
B. The axis of symmetry should be $x=2$.
C. The vertex should be a maximum.
D. The vertex should be $(-2,1)$.
Answer (1)
The vertex of the function f ( x ) = − ( x + 2 ) 2 − 1 is ( − 2 , − 1 ) .
The axis of symmetry is x = − 2 .
Since the coefficient of the x 2 term is negative, the vertex is a maximum.
Therefore, the correct statement is that the vertex should be a maximum. The vertex should be a maximum.
Explanation
Analyze the problem The given function is f ( x ) = − ( x + 2 ) 2 − 1 . We need to identify the error in the graph of this function based on the given options. The options relate to the axis of symmetry and the vertex of the parabola.
Find the vertex The vertex form of a parabola is given by f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In our case, f ( x ) = − ( x + 2 ) 2 − 1 , so we can identify a = − 1 , h = − 2 , and k = − 1 . Therefore, the vertex of the parabola is ( − 2 , − 1 ) .
Determine the axis of symmetry The axis of symmetry is a vertical line that passes through the vertex. The equation of the axis of symmetry is given by x = h . In our case, h = − 2 , so the axis of symmetry is x = − 2 .
Determine if the vertex is a maximum or minimum Since the coefficient a = − 1 is negative, the parabola opens downward. This means that the vertex is a maximum point.
Identify the error Now, let's compare our findings with the given options:
The axis of symmetry should be x = − 1 . This is incorrect; the axis of symmetry is x = − 2 .
The axis of symmetry should be x = 2 . This is incorrect; the axis of symmetry is x = − 2 .
The vertex should be a maximum. This is correct.
The vertex should be ( − 2 , 1 ) . This is incorrect; the vertex is ( − 2 , − 1 ) .
Therefore, the best description of the error in the graph is that the vertex should be a maximum.
Examples
Understanding the properties of quadratic functions, such as the vertex and axis of symmetry, is crucial in various real-world applications. For example, engineers use parabolas to design arches in bridges, ensuring structural stability. The vertex represents the highest or lowest point of the arch, which is critical for load distribution. Similarly, satellite dishes and reflectors use parabolic shapes to focus signals at the vertex, maximizing signal strength. By correctly identifying the vertex and axis of symmetry, engineers can optimize the design and performance of these structures.
