Which of the following equations has an axis of symmetry with the equation [tex]$x=3$[/tex]?
a. [tex]$y=x^2+6 x-3$[/tex]
c. [tex]$y=2 x^2+6 x-3$[/tex]
b. [tex]$y=x^2-6 x+3$[/tex]
d. [tex]$y=3 x^2+6 x-3$[/tex]
Answer (2)
The equation with an axis of symmetry at x = 3 is y = x 2 − 6 x + 3 , which corresponds to option (b).
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The axis of symmetry for a quadratic equation y = a x 2 + b x + c is given by x = − 2 a b .
Calculate the axis of symmetry for each equation.
Equation a: x = − 2 ( 1 ) 6 = − 3
Equation b: x = − 2 ( 1 ) − 6 = 3
Equation c: x = − 2 ( 2 ) 6 = − 2 3
Equation d: x = − 2 ( 3 ) 6 = − 1
The equation with the axis of symmetry x = 3 is y = x 2 − 6 x + 3 .
The final answer is B .
Explanation
Understanding the Axis of Symmetry The axis of symmetry of a quadratic equation in the form y = a x 2 + b x + c is given by the equation x = − 2 a b . We need to find the equation where − 2 a b = 3 .
Calculating the Axis of Symmetry for Each Equation Let's calculate the axis of symmetry for each of the given equations:
a. y = x 2 + 6 x − 3 : Here, a = 1 and b = 6 . So, the axis of symmetry is x = − 2 ( 1 ) 6 = − 3 .
b. y = x 2 − 6 x + 3 : Here, a = 1 and b = − 6 . So, the axis of symmetry is x = − 2 ( 1 ) − 6 = 3 .
c. y = 2 x 2 + 6 x − 3 : Here, a = 2 and b = 6 . So, the axis of symmetry is x = − 2 ( 2 ) 6 = − 2 3 .
d. y = 3 x 2 + 6 x − 3 : Here, a = 3 and b = 6 . So, the axis of symmetry is x = − 2 ( 3 ) 6 = − 1 .
Finding the Matching Equation Comparing the calculated axis of symmetry for each equation with the given axis of symmetry x = 3 , we find that equation b, y = x 2 − 6 x + 3 , has the axis of symmetry x = 3 .
Final Answer Therefore, the equation with an axis of symmetry at x = 3 is y = x 2 − 6 x + 3 . The answer is B.
Examples
Understanding the axis of symmetry is crucial in various real-world applications. For instance, designing bridges and arches requires understanding the symmetry to ensure structural stability and even weight distribution. Similarly, in physics, projectile motion follows a parabolic path, and the axis of symmetry helps determine the maximum height and range of the projectile. In business, understanding symmetry can help optimize marketing campaigns by identifying the point at which returns diminish, allowing for better resource allocation. The axis of symmetry is a powerful tool in mathematics and has far-reaching implications in engineering, physics, and business.
