Write the quadratic function in standard form.
[tex]f(x)=x^2+3 x+\frac{1}{4}[/tex]
[tex]f(x)=[/tex]
Sketch its graph.
Identify the vertex, axis of symmetry, and [tex]x[/tex]-intercept(s). (If an answer does not exist, enter DNE.)
vertex
[tex](x, f(x))=\left(-\frac{3}{2},-\frac{5}{4}{ }_x\right)[/tex]
axis of symmetry [tex]x=-\frac{3}{2}[/tex]
[tex]x[/tex]-intercept:
smaller [tex]x[/tex]-value [tex](x, f(x))=[/tex]
larger [tex]x[/tex]-value [tex](x, f(x))=[/tex]
Answer (1)
Express the quadratic function in standard form by completing the square: f ( x ) = ( x + 2 3 ) 2 − 2 .
Identify the vertex from the standard form: ( − 2 3 , − 2 ) .
Determine the axis of symmetry: x = − 2 3 .
Find the x-intercepts by setting f ( x ) = 0 and solving for x : x = − 2 3 ± 2 .
f ( x ) = ( x + 2 3 ) 2 − 2 ; ( − 2 3 , − 2 ) ; x = − 2 3 ; x = − 2 3 − 2 , − 2 3 + 2
Explanation
Problem Analysis We are given the quadratic function f ( x ) = x 2 + 3 x + 4 1 and we want to express it in standard form, sketch its graph, and identify its key features: vertex, axis of symmetry, and x-intercepts.
Standard Form The standard form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. To convert the given function to standard form, we complete the square.
Completing the Square Starting with f ( x ) = x 2 + 3 x + 4 1 , we complete the square for the x terms. We take half of the coefficient of the x term, which is 2 3 , and square it, which gives ( 2 3 ) 2 = 4 9 . We add and subtract this value within the expression:
f ( x ) = x 2 + 3 x + 4 9 − 4 9 + 4 1
Rewriting as a Square Now, we can rewrite the first three terms as a perfect square:
f ( x ) = ( x + 2 3 ) 2 − 4 9 + 4 1
Standard Form Result Combining the constants, we get:
f ( x ) = ( x + 2 3 ) 2 − 4 8 = ( x + 2 3 ) 2 − 2
So, the standard form of the quadratic function is f ( x ) = ( x + 2 3 ) 2 − 2 .
Vertex From the standard form f ( x ) = ( x + 2 3 ) 2 − 2 , we can identify the vertex as ( h , k ) = ( − 2 3 , − 2 ) .
Axis of Symmetry The axis of symmetry is a vertical line passing through the vertex, so its equation is x = − 2 3 .
Finding x-intercepts To find the x-intercepts, we set f ( x ) = 0 and solve for x :
( x + 2 3 ) 2 − 2 = 0
( x + 2 3 ) 2 = 2
x + 2 3 = ± 2
x = − 2 3 ± 2
x-intercepts Coordinates Thus, the x-intercepts are x = − 2 3 − 2 and x = − 2 3 + 2 . Approximating these values, we get x ≈ − 2.914 and x ≈ − 0.086 . The x-intercepts as coordinates are approximately ( − 2.914 , 0 ) and ( − 0.086 , 0 ) .
Final Summary In summary, the quadratic function in standard form is f ( x ) = ( x + 2 3 ) 2 − 2 . The vertex is ( − 2 3 , − 2 ) , the axis of symmetry is x = − 2 3 , and the x-intercepts are x = − 2 3 − 2 and x = − 2 3 + 2 .
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 range and maximum height of the projectile. Similarly, architects use quadratic functions to design parabolic arches in buildings and bridges, ensuring structural stability and aesthetic appeal. The vertex of the parabola represents the maximum or minimum point, which is essential in optimizing designs for efficiency and safety.
