To find the vertex and the axis of symmetry of the parabola given the points (-5, 0), (3, 0), and (-1, 16), we can follow these steps:
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Identify the x-coordinates of the roots: The points (-5, 0) and (3, 0) are the x-intercepts (roots) of the parabola.
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Calculate the axis of symmetry: The axis of symmetry of a parabola that opens upwards or downwards can be found by taking the average of the x-coordinates of its roots. The formula for the axis of symmetry (x) is: \[ x = \frac{x_1 + x_2}{2} \] where \(x_1\) and \(x_2\) are the x-coordinates of the roots. Here, the roots are -5 and 3. \[ x = \frac{-5 + 3}{2} = \frac{-2}{2} = -1 \] So, the axis of symmetry is \(x = -1\).
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Determine the vertex: The vertex lies on the axis of symmetry. To find the y-coordinate of the vertex, we can substitute \(x = -1\) into the equation of the parabola. First, we need to find the equation of the parabola.
To find the equation of the parabola, we can use the general form \(y = a(x - r_1)(x - r_2)\) where \(r_1 = -5\) and \(r_2 = 3\): \[ y = a(x + 5)(x - 3) \]
Now, we will use the point (-1, 16) to find the value of \(a\): \[ 16 = a(-1 + 5)(-1 - 3) \] \[ 16 = a(4)(-4) \] \[ 16 = -16a \] \[ a = -1 \]
Thus, the equation of the parabola is: \[ y = -1(x + 5)(x - 3) \] We can expand this: \[ y = -1(x^2 + 3x - 15) = -x^2 - 3x + 15 \]
Now, substitute \(x = -1\) to find the y-coordinate of the vertex: \[ y = -(-1)^2 - 3(-1) + 15 \] \[ y = -1 + 3 + 15 \] \[ y = 17 \]
- Conclusion: The vertex of the parabola is \((-1, 17)\) and the axis of symmetry is \(x = -1\).
So, the final results are:
- Vertex: \((-1, 17)\)
- Axis of Symmetry: \(x = -1\)
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