Write the quadratic function in standard form.
[tex]f(x)=x^2+3 x+\frac{1}{4}[/tex]
[tex]f(x)=[/tex]
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))=[/tex]
axis of symmetry
[tex]x=[/tex]
x-intercept:
smaller [tex]x[/tex]-value
[tex](x, f(x))=[/tex]
larger [tex]x[/tex]-value
[tex](x, f(x))=[/tex]
Answer (1)
Rewrite the quadratic function f ( x ) = x 2 + 3 x + 4 1 in standard form by completing the square: f ( x ) = ( x + 2 3 ) 2 − 2 .
Set f ( x ) = 0 to find the x-intercepts: ( x + 2 3 ) 2 − 2 = 0 .
Solve for x : x = − 2 3 ± 2 .
The x-intercepts are x = − 2 3 − 2 and x = − 2 3 + 2 , so the final answer is − 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, identify its vertex, axis of symmetry, and x-intercepts. The vertex and axis of symmetry are already provided, so we focus on rewriting the function in standard form and finding the x-intercepts.
Completing the Square The standard form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In our case, a = 1 . We complete the square to rewrite the given function in standard form:
f ( x ) = x 2 + 3 x + 4 1
To complete the square, we take half of the coefficient of the x term (which is 3), square it ( ( 2 3 ) 2 = 4 9 ), and add and subtract it within the expression:
f ( x ) = ( x 2 + 3 x + 4 9 ) − 4 9 + 4 1
Now, we can rewrite the expression in parentheses as a square:
f ( x ) = ( x + 2 3 ) 2 − 4 9 + 4 1
Simplify the constants:
f ( x ) = ( x + 2 3 ) 2 − 4 8
f ( x ) = ( x + 2 3 ) 2 − 2
So, the standard form of the quadratic function is f ( x ) = ( x + 2 3 ) 2 − 2 .
Finding the 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
Take the square root of both sides:
x + 2 3 = ± 2
Solve for x :
x = − 2 3 ± 2
Thus, the x-intercepts are x = − 2 3 − 2 and x = − 2 3 + 2 .
Approximate x-intercepts The x-intercepts are ( − 2 3 − 2 , 0 ) and ( − 2 3 + 2 , 0 ) . Approximating these values, we have x ≈ − 2.91 and x ≈ − 0.09 .
Final Answer In summary, the quadratic function in standard form is f ( x ) = ( x + 2 3 ) 2 − 2 . 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 bridges, ensuring structural stability and aesthetic appeal. The standard form of a quadratic function helps in easily identifying the vertex, which represents the maximum or minimum point, providing valuable information for optimization problems.
