Question

Identify theh vertex and axis of symmetry of the parabola with the given points: (-5, 0), (3, 0) (-1, 16)

Answer 1

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:

1. **Identify the x-coordinates of the roots**: The points (-5, 0) and (3, 0) are the x-intercepts (roots) of the parabola.

2. **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 \(x = -1\).

3. **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:
\[
y = -(-1)^2 - 3(-1) + 15
\]
\[
y = -1 + 3 + 15
\]
\[
y = 17
\]

4. **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: \(x = -1\)