To graph the piecewise function $f(x)$, we need to consider the function's definition over the three intervals specified:

1. For $x < -4$:
   $f(x) = 6$.

   • Here, the function is a horizontal line at $y = 6$ for all $x$ values less than $-4$.
   • There is no specific endpoint at $x = -4$ because $x = -4$ is not included in this interval.

2. For $-4 \leq x < 4$:
   $f(x) = -10 + x^2$.

   • This is a quadratic function (a parabola) that opens upwards.
   • The endpoint at $x = -4$ is included, so $f(-4) = -10 + (-4)^2 = -10 + 16 = 6$.
   • The endpoint at $x = 4$ is not included, so we will only evaluate the function at values close to but less than $x = 4$. For instance, $f(0) = -10 + 0^2 = -10$ and $f(3.9) = -10 + 3.9^2 \approx 5.21$.

3. For $x \geq 4$:
   $f(x) = 2x - 2$.

   • This is a linear function with a slope of 2 and a y-intercept of $-2$.
   • The endpoint at $x = 4$ is included, so $f(4) = 2(4) - 2 = 8 - 2 = 6$.

Summary of Key Points:

• The graph is a horizontal line at $y = 6$ for $x < -4$.
• The graph is a parabola $y = -10 + x^2$ between $x = -4$ and $x < 4$, starting at $( -4, 6 )$.
• The graph is a straight line with equation $y = 2x - 2$ for $x \geq 4$, starting from $(4, 6)$.

Points to Graph:

1. $(-4, 6)$ (included)
2. $(4, 6)$ (included)
3. Additional points for the parabola, e.g., $(0, -10)$, $(3.9, 5.21)$
4. For $x \geq 4$, the line can continue to points like $(5, 8)$, $(6, 10)$, etc.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you determine the vertex of the parabola in the piecewise function?
2. What happens to the continuity of the function at $x = -4$ and $x = 4$?
3. How would the graph change if the function for $x \geq 4$ was $2x + 2$ instead?
4. How does the derivative of $f(x)$ behave across the different intervals?
5. What would the graph look like if the quadratic portion was $x^2 - 10x$ instead?

Tip: When sketching piecewise functions, pay special attention to whether endpoints are included or excluded in each interval to ensure accurate graphing.