To graph the parabola defined by the equation:
[tex]\[ y = -2(x + 5)^2 + 1 \][/tex]
we start by finding the vertex and choosing specific points to plot on either side of the vertex. Here is a step-by-step solution for finding these points:
1. Identify the Vertex:
The vertex form of a parabola is given by [tex]\( y = a(x - h)^2 + k \)[/tex]. Comparing this with our equation [tex]\( y = -2(x + 5)^2 + 1 \)[/tex]:
- [tex]\( h = -5 \)[/tex]
- [tex]\( k = 1 \)[/tex]
Therefore, the vertex of the parabola is at [tex]\( (-5, 1) \)[/tex].
2. Calculate Points to the Left of the Vertex:
- For [tex]\( x = -6 \)[/tex]:
[tex]\[
y = -2(-6 + 5)^2 + 1 = -2(-1)^2 + 1 = -2(1) + 1 = -1
\][/tex]
Hence, the point is [tex]\( (-6, -1) \)[/tex].
- For [tex]\( x = -7 \)[/tex]:
[tex]\[
y = -2(-7 + 5)^2 + 1 = -2(-2)^2 + 1 = -2(4) + 1 = -7
\][/tex]
Hence, the point is [tex]\( (-7, -7) \)[/tex].
3. Calculate Points to the Right of the Vertex:
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
y = -2(-4 + 5)^2 + 1 = -2(1)^2 + 1 = -2(1) + 1 = -1
\][/tex]
Hence, the point is [tex]\( (-4, -1) \)[/tex].
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = -2(-3 + 5)^2 + 1 = -2(2)^2 + 1 = -2(4) + 1 = -7
\][/tex]
Hence, the point is [tex]\( (-3, -7) \)[/tex].
4. Summary of Points on the Parabola:
- Vertex: [tex]\( (-5, 1) \)[/tex]
- Points to the left: [tex]\( (-6, -1) \)[/tex], [tex]\( (-7, -7) \)[/tex]
- Points to the right: [tex]\( (-4, -1) \)[/tex], [tex]\( (-3, -7) \)[/tex]
These points provide a clear representation of the parabola's shape. When you plot these points on a graph, you’d connect them to visualize a downward-opening parabola centered on [tex]\( (-5, 1) \)[/tex]. The five points you need to plot are:
[tex]\[ (-5, 1), (-6, -1), (-7, -7), (-4, -1), (-3, -7) \][/tex]
Once you plot these points, you will observe the parabolic shape forming. The vertex is the highest point, and the parabola opens downwards because the coefficient of the squared term ([tex]\( -2 \)[/tex]) is negative.
