To graph the parabola [tex]\( y = (x + 1)^2 - 4 \)[/tex], follow these steps:
1. Identify the Vertex:
The equation [tex]\( y = (x + 1)^2 - 4 \)[/tex] is in vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex. Here, [tex]\( h = -1 \)[/tex] and [tex]\( k = -4 \)[/tex]. So, the vertex of the parabola is [tex]\((-1, -4)\)[/tex].
2. Calculate Points to the Left of the Vertex:
- Choose [tex]\( x = -3 \)[/tex]:
[tex]\[
y = ((-3) + 1)^2 - 4 = (-2)^2 - 4 = 4 - 4 = 0
\][/tex]
The point is [tex]\((-3, 0)\)[/tex].
- Choose [tex]\( x = -2 \)[/tex]:
[tex]\[
y = ((-2) + 1)^2 - 4 = (-1)^2 - 4 = 1 - 4 = -3
\][/tex]
The point is [tex]\((-2, -3)\)[/tex].
3. Calculate Points to the Right of the Vertex:
- Choose [tex]\( x = 0 \)[/tex]:
[tex]\[
y = (0 + 1)^2 - 4 = 1^2 - 4 = 1 - 4 = -3
\][/tex]
The point is [tex]\((0, -3)\)[/tex].
- Choose [tex]\( x = 1 \)[/tex]:
[tex]\[
y = (1 + 1)^2 - 4 = 2^2 - 4 = 4 - 4 = 0
\][/tex]
The point is [tex]\((1, 0)\)[/tex].
Combining all these points, we have:
- Vertex: [tex]\((-1, -4)\)[/tex]
- Points to the left of the vertex: [tex]\((-3, 0)\)[/tex] and [tex]\((-2, -3)\)[/tex]
- Points to the right of the vertex: [tex]\((0, -3)\)[/tex] and [tex]\((1, 0)\)[/tex]
Now, plot these points on a coordinate plane:
1. Plot [tex]\((-1, -4)\)[/tex].
2. Plot [tex]\((-3, 0)\)[/tex].
3. Plot [tex]\((-2, -3)\)[/tex].
4. Plot [tex]\((0, -3)\)[/tex].
5. Plot [tex]\((1, 0)\)[/tex].
After plotting these points, draw a smooth curve passing through them to complete the graph of the parabola. The parabola opens upwards with its vertex at [tex]\((-1, -4)\)[/tex], and these points should help you see the correct shape of the graph.
