Sanjay begins to correctly graph the function [tex]f(x)=(x+1)^2-3[/tex]. Based on the axis of symmetry and the vertex, which graph could be Sanjay's?
Answer (2)
The function is f ( x ) = ( x + 1 ) 2 − 3 .
The vertex form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where the vertex is ( h , k ) .
The vertex of the given function is ( − 1 , − 3 ) .
The axis of symmetry is x = − 1 .
Sanjay's graph must have a vertex at ( − 1 , − 3 ) and an axis of symmetry at x = − 1 .
Explanation
Identify the Function and Vertex Form The given function is f ( x ) = ( x + 1 ) 2 − 3 . This is a quadratic function in vertex form, which is f ( x ) = a ( x − h ) 2 + k , where the vertex of the parabola is at the point ( h , k ) and the axis of symmetry is the vertical line x = h .
Determine the Vertex In our case, we have f ( x ) = ( x + 1 ) 2 − 3 , which can be written as f ( x ) = 1 ( x − ( − 1 ) ) 2 + ( − 3 ) . Comparing this with the vertex form f ( x ) = a ( x − h ) 2 + k , we can identify the vertex as ( h , k ) = ( − 1 , − 3 ) .
Find the Axis of Symmetry The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is -1, the axis of symmetry is the line x = − 1 .
State the Vertex and Axis of Symmetry Therefore, the vertex of the parabola is ( − 1 , − 3 ) and the axis of symmetry is x = − 1 . Sanjay's graph must have these features.
Examples
Understanding the vertex and axis of symmetry of a parabola is useful in many real-world applications. For example, if you are designing a parabolic mirror for a telescope or a satellite dish, knowing the vertex and axis of symmetry helps you focus the incoming light or signal to a single point. Similarly, if you are modeling the trajectory of a projectile, the vertex represents the highest point the projectile will reach, and the axis of symmetry indicates that the projectile's path is symmetric around this point. These concepts are also used in optimization problems, such as finding the maximum or minimum value of a quadratic function, which can be applied to various fields like engineering, economics, and physics.
The vertex of the function f ( x ) = ( x + 1 ) 2 − 3 is at ( − 1 , − 3 ) and the axis of symmetry is x = − 1 . Sanjay's graph will show a U-shape curve opening upwards with these features. He should look for a graph that matches this vertex and symmetry line.
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