Find the axis of symmetry for the parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose y-coordinate is tho same. as the given point. f(x)=2(x+3)
2
−1:(−2,1) The axis of symmetry is (Type an equation Simply your answer)
The Correct Answer and Explanation is :
To find the axis of symmetry for the parabola given by the equation ( f(x) = 2(x + 3)^2 – 1 ), we can use the general form of a quadratic function, which is ( f(x) = a(x – h)^2 + k ). In this form, ((h, k)) is the vertex of the parabola, and the axis of symmetry can be represented by the vertical line ( x = h ).
Step 1: Identify the Vertex
The equation ( f(x) = 2(x + 3)^2 – 1 ) can be rewritten to identify ( h ) and ( k ):
- Here, ( a = 2 ), ( h = -3 ), and ( k = -1 ).
- Thus, the vertex of the parabola is at the point ((-3, -1)).
Step 2: Determine the Axis of Symmetry
The axis of symmetry for this parabola is the vertical line that passes through the vertex, given by the equation:
[
x = -3
]
Step 3: Find a Second Point
To find a second point on the parabola that has the same ( y )-coordinate as the given point ((-2, 1)), we first evaluate ( f(x) ) to find where ( f(x) = 1 ):
[
1 = 2(x + 3)^2 – 1
]
Step 4: Solve for ( x )
Rearranging the equation:
[
2(x + 3)^2 = 2 \implies (x + 3)^2 = 1
]
Taking the square root of both sides gives:
[
x + 3 = \pm 1
]
Step 5: Find the Possible Values for ( x )
- For ( x + 3 = 1 ):
[
x = 1 – 3 = -2
] - For ( x + 3 = -1 ):
[
x = -1 – 3 = -4
]
Thus, the two ( x )-values where the ( y )-coordinate is the same (1) are ( x = -2 ) and ( x = -4 ).
Conclusion
Therefore, the axis of symmetry is ( x = -3 ), and the second point on the parabola with the same ( y )-coordinate as ((-2, 1)) is ((-4, 1)). This shows the symmetry of the parabola around its axis, as the ( x )-values of the points are equidistant from the axis of symmetry.