Find the average rate of change of the function from x₁ to x₂. f(x) = x² + 2x from x₁ = 3 to x₂ = 5

Verified step by step guidance

1. Identify the function given: \(f(x) = x^{2} + 2x\).
2. Note the interval over which to find the average rate of change: from \(x_1 = 3\) to \(x_2 = 5\).
3. Calculate the function values at the endpoints: find \(f(3)\) and \(f(5)\) by substituting \(x = 3\) and \(x = 5\) into the function.
4. Use the formula for average rate of change: \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\).
5. Substitute the values of \(f(5)\), \(f(3)\), \(x_2\), and \(x_1\) into the formula and simplify the expression to find the average rate of change.

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. It is calculated as the difference in function values divided by the difference in input values, similar to the slope of a secant line between two points on the graph.

Function Evaluation

Function evaluation involves substituting specific input values into the function to find corresponding output values. For example, to find f(3) and f(5) for f(x) = x² + 2x, substitute 3 and 5 into the expression and simplify to get the function values needed for the average rate of change.

Quadratic Functions

A quadratic function is a polynomial of degree two, typically in the form f(x) = ax² + bx + c. Its graph is a parabola, and understanding its shape helps interpret changes in function values. Here, f(x) = x² + 2x is quadratic, so its rate of change varies across intervals.