Math Problem Statement

For y=x-6/x+5, find the average rate of change from -4 ≤x ≤-1.

Solution

To find the average rate of change of y=x6x+5y = \frac{x - 6}{x + 5} over the interval [4,1][-4, -1], we use the formula:

Average Rate of Change=y2y1x2x1,\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}, where x1=4x_1 = -4, x2=1x_2 = -1, and y1,y2y_1, y_2 are the values of yy at x1x_1 and x2x_2, respectively.


Step 1: Compute y1y_1 at x=4x = -4

y1=x6x+5=464+5=101=10.y_1 = \frac{x - 6}{x + 5} = \frac{-4 - 6}{-4 + 5} = \frac{-10}{1} = -10.

Step 2: Compute y2y_2 at x=1x = -1

y2=x6x+5=161+5=74.y_2 = \frac{x - 6}{x + 5} = \frac{-1 - 6}{-1 + 5} = \frac{-7}{4}.


Step 3: Calculate the average rate of change

Substitute y1=10y_1 = -10, y2=74y_2 = -\frac{7}{4}, x1=4x_1 = -4, and x2=1x_2 = -1 into the formula:

Average Rate of Change=y2y1x2x1.\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}.

Average Rate of Change=74(10)1(4).\text{Average Rate of Change} = \frac{-\frac{7}{4} - (-10)}{-1 - (-4)}.

Simplify the numerator and denominator: Numerator: 74+10=74+404=334.\text{Numerator: } -\frac{7}{4} + 10 = -\frac{7}{4} + \frac{40}{4} = \frac{33}{4}.

Denominator: 1+4=3.\text{Denominator: } -1 + 4 = 3.

Average Rate of Change=3343=3343=3312.\text{Average Rate of Change} = \frac{\frac{33}{4}}{3} = \frac{33}{4 \cdot 3} = \frac{33}{12}.

Simplify the fraction: 3312=114.\frac{33}{12} = \frac{11}{4}.


Final Answer:

The average rate of change is: 114\boxed{\frac{11}{4}}

Would you like a deeper breakdown or further assistance?


Related Questions:

  1. How do you interpret the average rate of change in this context?
  2. What happens if x=5x = -5, where the denominator becomes zero?
  3. How would the interval change the result if it were [3,1][-3, -1]?
  4. Can you derive a general formula for the average rate of change of this function over any interval?
  5. What is the significance of the slope being positive, negative, or zero in this scenario?

Tip:

Always check for points where the denominator might become zero when working with rational functions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Average Rate of Change

Formulas

Average Rate of Change = (y2 - y1) / (x2 - x1)

Theorems

Properties of Rational Functions

Suitable Grade Level

Grades 10-12