From the equation, find the axis of symmetry of the parabola. [tex]y=4 x^2+32 x+61[/tex]
A. [tex]x=3[/tex]
B. [tex]x=4[/tex]
C. [tex]x=-4[/tex]
D. [tex]x=-3[/tex]

The axis of symmetry for the parabola given by the equation y = 4 x 2 + 32 x + 61 is x = − 4 . This was found using the formula x = − 2 a b ​ by identifying a and b . Therefore, the correct choice is option C: x = − 4 .
;

Answered by Anonymous | 2025-07-04

Identify the coefficients a and b from the quadratic equation y = a x 2 + b x + c .
Apply the formula for the axis of symmetry: x = − 2 a b ​ .
Substitute the values a = 4 and b = 32 into the formula.
Calculate the axis of symmetry: x = − 4 . The final answer is x = − 4 ​

Explanation

Understanding the Problem We are given the equation of a parabola y = 4 x 2 + 32 x + 61 and asked to find its axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves.

Identifying Coefficients and the Formula The general form of a quadratic equation is y = a x 2 + b x + c , where a , b , and c are constants. The axis of symmetry for a parabola in this form is given by the formula: x = − 2 a b ​ In our equation, y = 4 x 2 + 32 x + 61 , we can identify the coefficients as a = 4 , b = 32 , and c = 61 .

Calculating the Axis of Symmetry Now, we substitute the values of a and b into the formula for the axis of symmetry: x = − 2 ( 4 ) 32 ​ x = − 8 32 ​ x = − 4

Final Answer Therefore, the axis of symmetry of the parabola y = 4 x 2 + 32 x + 61 is x = − 4 .


Examples
Understanding the axis of symmetry is crucial in various real-world applications. For instance, when designing parabolic reflectors for satellite dishes or solar ovens, knowing the axis of symmetry helps in focusing signals or heat efficiently. Similarly, in architecture, the axis of symmetry can guide the design of arches and suspension bridges to ensure structural balance and aesthetic appeal. By understanding the properties of parabolas, engineers and designers can optimize their creations for maximum performance and stability.

Answered by GinnyAnswer | 2025-07-04