Which is the graph of the function [tex]f(x)=x^2+2 x+3[/tex] ?
Answer (1)
Determine that the parabola opens upwards since the coefficient of x 2 is positive.
Calculate the vertex of the parabola using x = − 2 a b and f ( x ) , which gives ( − 1 , 2 ) .
Find the y-intercept by setting x = 0 , resulting in ( 0 , 3 ) .
Identify the graph based on the upward opening, vertex, and y-intercept.
Explanation
Analyze the quadratic function We are given the quadratic function f ( x ) = x 2 + 2 x + 3 and asked to identify its graph. Since the coefficient of the x 2 term is positive, we know the parabola opens upwards. We need to find the vertex and y-intercept to determine the correct graph.
Find the vertex To find the vertex, we can complete the square or use the formula x = − 2 a b for the x-coordinate of the vertex. In this case, a = 1 and b = 2 , so the x-coordinate of the vertex is x = − 2 ( 1 ) 2 = − 1 . To find the y-coordinate of the vertex, we plug x = − 1 into the function: f ( − 1 ) = ( − 1 ) 2 + 2 ( − 1 ) + 3 = 1 − 2 + 3 = 2 . Therefore, the vertex is ( − 1 , 2 ) .
Find the y-intercept To find the y-intercept, we set x = 0 in the function: f ( 0 ) = ( 0 ) 2 + 2 ( 0 ) + 3 = 3 . So the y-intercept is ( 0 , 3 ) .
Identify the graph Now we know the parabola opens upwards, has a vertex at ( − 1 , 2 ) , and a y-intercept at ( 0 , 3 ) . This information should be enough to identify the correct graph.
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By knowing the initial velocity and launch angle, they can predict the maximum height and range of the projectile. Similarly, architects use quadratic functions to design arches and bridges, ensuring structural stability and optimal aesthetics. The vertex of the parabola represents the maximum or minimum point, which is essential in optimizing designs and performance.
