find the maximum value of y = −5x + 4 over the interval −4 ≤ x ≤ 3.

To find the maximum value of the function \( y = -5x + 4 \) over the interval \( -4 \leq x \leq 3 \), we need to evaluate the function at the endpoints of the interval since the function is linear.

1. **Evaluate at \( x = -4 \):**

   \[
   y = -5(-4) + 4 = 20 + 4 = 24
   \]

2. **Evaluate at \( x = 3 \):**

   \[
   y = -5(3) + 4 = -15 + 4 = -11
   \]

The maximum value of \( y \) over the interval occurs at \( x = -4 \) and is \( y = 24 \).

**Summary**:  
The maximum value of \( y = -5x + 4 \) over the interval \( -4 \leq x \leq 3 \) is \( 24 \), which occurs at \( x = -4 \).

Would you like more details or have any questions about this? Here are some related questions:

1. How do you determine the minimum value of a linear function over a given interval?
2. What is the significance of evaluating the function at the endpoints of the interval?
3. How does the slope of the line affect the location of the maximum and minimum values?
4. How would the result change if the interval were different?
5. Can you explain how the function behaves if the coefficient of \( x \) were positive instead of negative?

**Tip**: For linear functions, the maximum and minimum values on a closed interval always occur at the endpoints of the interval.

Learn how to find the maximum value of the linear function y = -5x + 4 over the interval [-4, 3]. Includes step-by-step evaluation at interval endpoints and understanding the behavior of linear functions.

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