Suppose that the function fis defined for all real numbers as follows. f(x) = 6 if x < - 4; - 10 + x ^ 2 if - 4 <= x < 4; 2x - 2 if x >= 4  Find the points to graph

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.

Explore how to graph piecewise functions step-by-step with an example including the function f(x) defined over different intervals. Includes points to graph and key details on continuity and behavior at specific points. Ideal for high school students learning piecewise functions.

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