Algebra

Find the average rate of change of f(x) = x^2 - 3x + 7 from x_1 = 1 to x_2 = 5.

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 over an interval. It's the slope of the secant line between two points on the curve.
2

Evaluate f(1) at x₁ = 1

We plug x = 1 into the function to find the output at the left endpoint of the interval.
3

Evaluate f(5) at x₂ = 5

We plug x = 5 into the function to find the output at the right endpoint of the interval.
4

Substitute values into the formula

Now we drop our computed function values and the x-values into the formula. The denominator is the change in x, and the numerator is the change in output.
5

Simplify to get the final answer

Dividing 12 by 4 gives us 3. This means on average, the function rises 3 units for every 1 unit increase in x across this interval.

Understanding this problem

Learning Insight

The average rate of change is really just slope — rise over run — but applied to a curve instead of a straight line. For a parabola like , the rate of change is different at every point, so the 'average' gives you the slope of the straight line connecting the two endpoints on the curve. In calculus, this idea leads directly to the derivative.

Quick Tip

To quickly check your function evaluations, write out each term separately before combining: , then , then . Adding step by step reduces arithmetic errors much more than trying to do it all at once.

Common Mistake

Students often subtract the x-values in the wrong order — writing on top but on the bottom (or vice versa). Always keep the same order in both the numerator and denominator: 'second minus first' on both.