Graph the function.
$g(x)=x^2+4 x+1$
Answer (1)
Find the vertex of the parabola by completing the square or using the formula x = − b / ( 2 a ) , resulting in ( − 2 , − 3 ) .
Determine the axis of symmetry, which is a vertical line passing through the vertex, x = − 2 .
Calculate the y-intercept by setting x = 0 , giving the point ( 0 , 1 ) .
Find the x-intercepts by setting g ( x ) = 0 and using the quadratic formula, resulting in x = − 2 ± 3 .
The graph is a parabola with vertex ( − 2 , − 3 ) , axis of symmetry x = − 2 , y-intercept ( 0 , 1 ) , and x-intercepts ( − 2 + 3 , 0 ) and ( − 2 − 3 , 0 ) .
g ( x ) = x 2 + 4 x + 1
Explanation
Understanding the Problem We are given the quadratic function g ( x ) = x 2 + 4 x + 1 and asked to graph it. To do this, we'll find the vertex, axis of symmetry, y-intercept, and x-intercepts. This will give us enough information to sketch the parabola.
Finding the Vertex First, let's find the vertex of the parabola. We can complete the square to rewrite the function in vertex form, g ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. Alternatively, we can use the formula x = − b / ( 2 a ) to find the x-coordinate of the vertex. In this case, a = 1 and b = 4 , so x = − 4/ ( 2 ∗ 1 ) = − 2 . Now, we can find the y-coordinate of the vertex by plugging x = − 2 into the function: g ( − 2 ) = ( − 2 ) 2 + 4 ( − 2 ) + 1 = 4 − 8 + 1 = − 3 . Thus, the vertex is at ( − 2 , − 3 ) .
Finding 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 -2, the axis of symmetry is x = − 2 .
Finding the y-intercept To find the y-intercept, we set x = 0 : g ( 0 ) = ( 0 ) 2 + 4 ( 0 ) + 1 = 1 . So, the y-intercept is at ( 0 , 1 ) .
Finding the x-intercepts To find the x-intercepts, we set g ( x ) = 0 : x 2 + 4 x + 1 = 0 . We can use the quadratic formula to solve for x: x = 2 a − b ± b 2 − 4 a c = 2 ( 1 ) − 4 ± 4 2 − 4 ( 1 ) ( 1 ) = 2 − 4 ± 16 − 4 = 2 − 4 ± 12 = 2 − 4 ± 2 3 = − 2 ± 3 . So, the x-intercepts are at ( − 2 + 3 , 0 ) and ( − 2 − 3 , 0 ) . Since 3 ≈ 1.732 , the x-intercepts are approximately at ( − 0.268 , 0 ) and ( − 3.732 , 0 ) .
Sketching the Parabola Now we have the vertex ( − 2 , − 3 ) , the axis of symmetry x = − 2 , the y-intercept ( 0 , 1 ) , and the x-intercepts ( − 2 + 3 , 0 ) and ( − 2 − 3 , 0 ) . We can plot these points and sketch the parabola. The parabola opens upwards since the coefficient of the x 2 term is positive.
Final Answer The graph of the function g ( x ) = x 2 + 4 x + 1 is a parabola with vertex at ( − 2 , − 3 ) , axis of symmetry x = − 2 , y-intercept at ( 0 , 1 ) , and x-intercepts at ( − 2 + 3 , 0 ) and ( − 2 − 3 , 0 ) .
Examples
Understanding quadratic functions like g ( x ) = x 2 + 4 x + 1 is crucial in various real-world applications. For instance, if you're launching a projectile, the height of the projectile over time can be modeled by a quadratic function. By finding the vertex, you can determine the maximum height the projectile reaches. Similarly, businesses use quadratic functions to model profit curves, where the vertex indicates the point of maximum profit. Graphing these functions helps visualize the behavior and make informed decisions.
