To graph the equation [tex]\( y = (x - 6)^2 + 2 \)[/tex], follow these steps:
### Step 1: Understand the Equation
The given equation [tex]\( y = (x - 6)^2 + 2 \)[/tex] is a quadratic function. The general form of a quadratic function is [tex]\( y = a(x - h)^2 + k \)[/tex], which represents a parabola. Here, [tex]\( a = 1 \)[/tex], [tex]\( h = 6 \)[/tex], and [tex]\( k = 2 \)[/tex].
- The vertex of the parabola is at the point [tex]\( (h, k) = (6, 2) \)[/tex].
- Since [tex]\( a \)[/tex] is positive, the parabola opens upwards.
### Step 2: Identify Key Points
To graph the function accurately, identify several key points around the vertex.
1. Vertex: The vertex is at [tex]\( (6, 2) \)[/tex].
2. Axis of Symmetry: The line [tex]\( x = 6 \)[/tex] is the axis of symmetry of the parabola.
3. Calculate the value of [tex]\( y \)[/tex] for various [tex]\( x \)[/tex]-values around the vertex:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = (0 - 6)^2 + 2 = 36 + 2 = 38 \)[/tex]
- When [tex]\( x = 4 \)[/tex]: [tex]\( y = (4 - 6)^2 + 2 = 4 + 2 = 6 \)[/tex]
- When [tex]\( x = 5 \)[/tex]: [tex]\( y = (5 - 6)^2 + 2 = 1 + 2 = 3 \)[/tex]
- When [tex]\( x = 7 \)[/tex]: [tex]\( y = (7 - 6)^2 + 2 = 1 + 2 = 3 \)[/tex]
- When [tex]\( x = 8 \)[/tex]: [tex]\( y = (8 - 6)^2 + 2 = 4 + 2 = 6 \)[/tex]
- When [tex]\( x = 12 \)[/tex]: [tex]\( y = (12 - 6)^2 + 2 = 36 + 2 = 38 \)[/tex]
### Step 3: Plot the Points
Now, plot the points on a Cartesian plane:
- [tex]\( (0, 38) \)[/tex]
- [tex]\( (4, 6) \)[/tex]
- [tex]\( (5, 3) \)[/tex]
- [tex]\( (6, 2) \)[/tex] (vertex)
- [tex]\( (7, 3) \)[/tex]
- [tex]\( (8, 6) \)[/tex]
- [tex]\( (12, 38) \)[/tex]
### Step 4: Draw the Parabola
Using the points plotted, sketch the parabola. Remember that the parabola should be symmetric about the vertical line [tex]\( x = 6 \)[/tex].
### Step 5: Add Details
- Label the vertex at [tex]\( (6, 2) \)[/tex].
- Draw and label the axis of symmetry at [tex]\( x = 6 \)[/tex].
- Include a title such as "Graph of [tex]\( y = (x - 6)^2 + 2 \)[/tex]".
- Label the x-axis and y-axis appropriately.
### Graph Summary
The parabola is characterized by:
- A vertex at [tex]\( (6, 2) \)[/tex]
- Opens upwards because the coefficient of [tex]\((x - 6)^2\)[/tex] is positive
- Symmetry about the line [tex]\( x = 6 \)[/tex]
By following these steps, you should have an accurate graph of the equation [tex]\( y = (x - 6)^2 + 2 \)[/tex].
