To graph the quadratic function $f(x) = (x + 6)^2 + 2$, you'll follow these steps based on the transformation of the basic quadratic function $y = x^2$:

1. Identify the Transformations:

• Horizontal Shift: The $x + 6$ inside the squared term indicates a horizontal shift. Specifically, the graph of $y = x^2$ will shift 6 units to the left.
• Vertical Shift: The $+2$ outside the squared term indicates a vertical shift. The graph will move 2 units up.

2. Determine the Vertex:

• The vertex of the basic function $y = x^2$ is at the origin $(0, 0)$.
• After the transformations, the vertex of $f(x) = (x + 6)^2 + 2$ will be shifted to the point $(-6, 2)$.

3. Plot the Vertex:

• Start by plotting the vertex at $(-6, 2)$.

4. Plot Additional Points:

• Choose a few x-values around the vertex and calculate the corresponding y-values.
• For example:
  • If $x = -7$: $f(-7) = (-7 + 6)^2 + 2 = 1^2 + 2 = 3$.
  • If $x = -5$: $f(-5) = (-5 + 6)^2 + 2 = 1^2 + 2 = 3$.
  • If $x = -4$: $f(-4) = (-4 + 6)^2 + 2 = 2^2 + 2 = 6$.

5. Sketch the Parabola:

• Connect the points with a smooth curve, ensuring that it opens upwards, as the coefficient of $x^2$ is positive.

6. Graph the Function:

• Use the graphing tool to plot the points and draw the curve.

Graph Summary:

• Vertex: $(-6, 2)$
• Direction: Opens upwards.
• Shifts: Left 6 units, up 2 units.

Would you like more details or have any questions?

Follow-up Questions:

1. How would the graph change if the equation was $f(x) = (x - 6)^2 + 2$?
2. What is the axis of symmetry for the function $f(x) = (x + 6)^2 + 2$?
3. How does the coefficient of the squared term affect the graph's shape?
4. What are the key differences between the graph of $f(x) = x^2$ and $f(x) = (x + 6)^2 + 2$?
5. Can you derive the function's formula from a different vertex and transformation?

Tip: Always start by identifying the vertex and direction of the parabola when graphing quadratic functions.