Sure! Let's graph the given parabola [tex]\( y = (x-4)^2 - 1 \)[/tex] and identify five key points including the vertex and four additional points.
### Step-by-Step Solution:
1. Understanding the Equation:
- The given equation is [tex]\( y = (x-4)^2 - 1 \)[/tex].
- This is a standard form of a parabola [tex]\( y = a(x-h)^2 + k \)[/tex] where the vertex is at the point [tex]\((h, k)\)[/tex].
2. Identify the Vertex:
- By comparing [tex]\( y = (x-4)^2 - 1 \)[/tex] with [tex]\( y = a(x-h)^2 + k \)[/tex]:
- [tex]\( h = 4 \)[/tex]
- [tex]\( k = -1 \)[/tex]
- So, the vertex of the parabola is at [tex]\((4, -1)\)[/tex].
3. Choose Additional Points:
- We will pick points around the vertex to keep it simple, say [tex]\( x = 3, 5, 2, \)[/tex] and [tex]\( 6 \)[/tex].
4. Calculate y-values for Chosen x-values:
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = (4-4)^2 - 1 = 0 - 1 = -1
\][/tex]
[tex]\((4, -1)\)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = (3-4)^2 - 1 = 1^2 - 1 = 1 - 1 = 0
\][/tex]
[tex]\((3, 0)\)[/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
y = (5-4)^2 - 1 = 1^2 - 1 = 1 - 1 = 0
\][/tex]
[tex]\((5, 0)\)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = (2-4)^2 - 1 = 2^2 - 1 = 4 - 1 = 3
\][/tex]
[tex]\((2, 3)\)[/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = (6-4)^2 - 1 = 2^2 - 1 = 4 - 1 = 3
\][/tex]
[tex]\((6, 3)\)[/tex]
5. Plot the Points and the Parabola:
- Now we plot the points [tex]\((4, -1)\)[/tex], [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex].
- Also, we draw the curve of the parabola [tex]\( y = (x-4)^2 - 1 \)[/tex].
### Graphical Representation:
- Vertex: The parabola’s vertex is [tex]\( (4, -1) \)[/tex].
- Points:
- [tex]\( (4, -1) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
- [tex]\( (5, 0) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
- [tex]\( (6, 3) \)[/tex]
Here's how you'd sketch the graph on Cartesian coordinates:
- Plot the vertex at [tex]\((4, -1)\)[/tex].
- Plot the other points: [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex].
- The parabola opens upwards because the coefficient of [tex]\((x-4)^2\)[/tex] is positive.
- Draw a smooth curve through these points to complete the parabola.
### Summary
- Vertex of the parabola: [tex]\((4, -1)\)[/tex]
- Additional Points: [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex]
- The equation of the parabola is [tex]\( y = (x-4)^2 - 1 \)[/tex].
This should guide you in graphing the parabola with the specified points.
