graph 2x+5 if x less than or equal to 4        5 if -4<x=<4           5x-4 if x > 4

To graph the piecewise function, let's break it down into three parts based on the given conditions:

1. **For \( x \leq 4 \):** The function is \( 2x + 5 \).
2. **For \( -4 < x \leq 4 \):** The function is a constant \( 5 \).
3. **For \( x > 4 \):** The function is \( 5x - 4 \).

Now, let's summarize how each section behaves:

### 1. **Graph for \( 2x + 5 \) when \( x \leq 4 \):**
   - This is a straight line with slope 2 and y-intercept 5.
   - It will include the point at \( x = 4 \) and go towards negative infinity for \( x < 4 \).
   - At \( x = 4 \), \( y = 2(4) + 5 = 13 \), so one point on the graph is \( (4, 13) \).

### 2. **Graph for \( 5 \) when \( -4 < x \leq 4 \):**
   - This is a horizontal line at \( y = 5 \), starting just after \( x = -4 \) and continuing until \( x = 4 \).
   - At \( x = 4 \), the function value is 5.

### 3. **Graph for \( 5x - 4 \) when \( x > 4 \):**
   - This is another straight line with slope 5 and y-intercept -4.
   - This line begins just after \( x = 4 \) and continues as \( x \) increases.
   - At \( x = 4 \), the value is \( 5(4) - 4 = 16 \), but since \( x = 4 \) is not included in this part of the piecewise function, the graph starts from a point just after \( (4, 16) \).

Would you like me to plot this piecewise function for you?

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Here are five related questions to expand your understanding:
1. What is the slope of each piece of the function?
2. How do discontinuities in piecewise functions affect the graph?
3. What are some real-world applications of piecewise functions?
4. Can you explain how to find the intercepts for each part of the function?
5. How does the behavior of each function change at their respective boundaries?

**Tip:** When graphing piecewise functions, always check for open and closed circles to indicate whether the endpoints are included.

graph 2x+5 if x less than or equal to 4 5 if -4<x=<4 5x-4 if x > 4

This lesson explores how to graph the piecewise function 2x+5 for x ≤ 4, 5 for -4 < x ≤ 4, and 5x-4 for x > 4. Includes step-by-step breakdowns, identification of key points, and graphical behavior at boundaries. Ideal for high school students in Grades 9-12.

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