Algebra

Find the average rate of change of the function from x_1 to x_2. f(x) = -2x + 15, x_1 = 0, x_2 = 3

Step-by-step solution with explanation

Final Answer

The average rate of change is .

Step-by-step solution

1

Write the average rate of change formula

The average rate of change measures how much the function's output changes per unit of input. It's just the slope formula between two points.
2

Find f(x_1) by plugging in x_1 = 0

We substitute 0 into the function to get the output value at the first point.
3

Find f(x_2) by plugging in x_2 = 3

We substitute 3 into the function to get the output value at the second point.
4

Plug values into the formula

We place the two output values in the numerator and the two input values in the denominator, keeping the order consistent.
5

Simplify the fraction to get the answer

Dividing gives us the average rate of change. This tells us the function drops by 2 for every 1 unit increase in x.

Understanding this problem

Learning Insight

For any linear function like , the average rate of change between ANY two points is always equal to the slope (the coefficient of x). That's what makes linear functions special — their rate of change is constant everywhere.

Quick Tip

For a linear function , the average rate of change is always just — no calculation needed! Here, , so the answer is .

Common Mistake

Students often subtract in the wrong order — for example, computing in the numerator but in the denominator. Always keep the same order: second minus first, top and bottom.