Which graph could represent the function [tex]f(x)=(x+3.8)^2-2.7[/tex]?
Answer (1)
The given function is a quadratic function in vertex form.
The vertex of the parabola is at ( − 3.8 , − 2.7 ) .
Since the coefficient of the x 2 term is positive, the parabola opens upwards.
The graph is a parabola opening upwards with vertex at ( − 3.8 , − 2.7 ) .
Explanation
Identifying the Function Type and Vertex Form We are given the function f ( x ) = ( x + 3.8 ) 2 − 2.7 . This is a quadratic function in vertex form, which is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In our case, a = 1 , h = − 3.8 , and k = − 2.7 .
Determining the Vertex and Direction The vertex of the parabola is at ( h , k ) = ( − 3.8 , − 2.7 ) . Since 0"> a = 1 > 0 , the parabola opens upwards. This means the vertex is the minimum point of the graph.
Conclusion about the Graph Therefore, the graph of the function f ( x ) = ( x + 3.8 ) 2 − 2.7 is a parabola that opens upwards with its vertex at ( − 3.8 , − 2.7 ) .
Examples
Understanding quadratic functions is crucial in various fields, such as physics and engineering. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic function. If you launch a ball, the height of the ball over time follows a parabolic path. The vertex of this parabola represents the maximum height the ball reaches. By analyzing the quadratic function, you can determine the launch angle and initial velocity needed to hit a target at a specific distance.
