Certainly! Let's graph the parabola defined by the equation [tex]\( y = x^2 - 6 \)[/tex]. Here is a detailed, step-by-step approach to graphing it:
### Step 1: Understanding the Equation
The equation [tex]\( y = x^2 - 6 \)[/tex] represents a parabola. This parabola opens upwards because the coefficient of [tex]\( x^2 \)[/tex] is positive.
### Step 2: Determine Key Points
To graph the parabola, it's helpful to determine some key points, including the vertex, the axis of symmetry, and a few additional points:
#### Vertex
The standard form of a parabola is [tex]\( y = a(x - h)^2 + k \)[/tex]. In this equation:
- [tex]\( a = 1 \)[/tex]
- [tex]\( h = 0 \)[/tex] (since there is no [tex]\( x \)[/tex] term inside the parentheses)
- [tex]\( k = -6 \)[/tex]
So, the vertex of the parabola is at (0, -6).
#### Axis of Symmetry
The axis of symmetry for a parabola [tex]\( y = a(x - h)^2 + k \)[/tex] is the vertical line [tex]\( x = h \)[/tex]. In this case, the axis of symmetry is [tex]\( x = 0 \)[/tex].
#### Additional Points
Choose values for [tex]\( x \)[/tex] around the vertex to find corresponding [tex]\( y \)[/tex] values:
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = (-2)^2 - 6 = 4 - 6 = -2 \)[/tex]
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = (-1)^2 - 6 = 1 - 6 = -5 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = 1^2 - 6 = 1 - 6 = -5 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( y = 2^2 - 6 = 4 - 6 = -2 \)[/tex]
### Step 3: Plot the Points
Now, plot the following points on the coordinate plane:
- Vertex: (0, -6)
- Additional points: (-2, -2), (-1, -5), (1, -5), (2, -2)
### Step 4: Draw the Parabola
1. Start at the vertex (0, -6).
2. Plot the additional points.
3. Draw a smooth curve through these points to form the parabola, ensuring it opens upward.
### Step 5: Confirm Parabola Symmetry
Ensure that your graph is symmetrical about the axis of symmetry [tex]\( x = 0 \)[/tex].
### Visualization
To help visualize, you could either plot this on graph paper or imagine the following points connected smoothly:
```
y
^
|
| | |
-4|
-5|
-6|_______________*____ x
-2 -1 0 1 2
```
The above steps will allow you to graph the parabola [tex]\( y = x^2 - 6 \)[/tex] accurately.
